NASA Contractor Report 201686

An Approach to the Constrained Designof Natural Laminar Flow Airfoils

Bradford E. Green

The George Washington University

Joint Institute for Advancement of Flight Sciences

Langley Research Center, Hampton, Virginia

Cooperative Agreement NCC1-24

February 1997

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-0001

https://ntrs.nasa.gov/search.jsp?R=19970019686 2018-06-24T13:50:41+00:00Z

• ._ . ,

Table of Contents

List of Tables ...................................................................................................................... iii

List of Figures .................................................................................................................... iv

List of Symbols ................................................................................................................. vii

Abstract ................................................................................................................................ 1

1.0 Introduction ................................................................................................................. 1

2.0 Overview of the Airfoil Design Method ..................................................................... 2

2.1 Euler Solver ....................................................................................................... 3

2.2 Turbulent Boundary Layer Method ................................................................... 4

2.3 Laminar Boundary Layer Solver ....................................................................... 5

2.4 Stability Analysis Code ..................................................................................... 6

2.4.1 Calculation of Analysis N-Factor Distribution ...................................... 7

2.4.2 Use of N-Factors to Estimate the Transition Location .......................... 8

2.5 Airfoil Design Method ....................................................................................... 9

3.0 Obtaining the Target Pressure Distribution ................................................................. 9

3.1 The Target N-Factor Distribution .................................................................... 10

3.1.1 Subcritical Cases .................................................................................. 10

3.1.2 Supercritical Cases ............................................................................... 11

3.2 Extrapolation of Analysis N-Factors ............................................................... 12

3.3 The Upper Surface Target Pressures ................................................................ 14

3.3.1 The N-Factor Design Method .............................................................. 14

3.3.2 The Pressure Recovery Region ............................................................ 16

3.4 Meeting the Aerodynamic Constraints ............................................................ 18

3.4.1 Lift Coefficient ..................................................................................... 19

3.4.2 Pitching Moment Coefficient ............................................................... 20

3.5 Adjusting the Leading-edge Radius ................................................................. 22

3.6 The Lower Surface Target Pressures ............................................................... 22

4.0 Modifying Target Pressures to Enforce Constraints ................................................. 23

4.1 Leading-Edge Pressures ................................................................................... 24

4.2 Trailing-Edge Pressures ................................................................................... 24

4.3 Releasing Constraints ....................................................................................... 25

5.0 Results of the NLF Airfoil Design Method .............................................................. 27

5.1 Airfoil for a Glider ........................................................................................... 27

5.2 Airfoil for a Commuter Aircraft ...................................................................... 28

5.3 Airfoil for a Subsonic Transport Aircraft ........................................................ 31

6.0 Concluding Remarks ................................................................................................. 33

7.0 Acknowledgments ..................................................................................................... 34

8.0 References ........................................................................... . ..................................... 34

Appendix A. A Linear Scaling Method ............................................................................ 65

Appendix B. A Weighted Averaging Technique ............................................................... 67

ii

List of Tables

o A comparison of the design constraints and the characteristics of the NACA 641-

212 and the redesigned airfoil at M_ = 0.10, Re = 3 million, and c1 = 0.30 ............. 28

, A comparison of the design constraints and the characteristics of the NACA 1412

and the redesigned airfoil at M_ = 0.60, Re = 20 million, and c l = 0.40 .................. 30

o A comparison of the design constraints and the characteristics of the NASA High

Speed NLF-0213 and the two redesigned airfoils at M_ = 0.60, Re = 20 million,

and c l = 0.40 .............................................................................................................. 31

, A comparison of the design constraints and the characteristics of the NASA

Supercritical SC(2)-0412 airfoil and the redesigned airfoil at M_ = 0.76, Re = 10

million, and c l -- 0.50 ................................................................................................ 32

iii

List of Figures

1. A flowchart of the NLF airfoil design method .......................................................... 37

2. A typical variation of N-Factor (N) with frequency (C0r) at any x-location on an

airfoil for a constant wave angle (_) ........................................................................ 38

3. A flowchart of how COSAL is used to calculate the analysis N-Factor distribution

for a pressure distribution ......................................................................................... 39

4. A flowchart of the Target Pressure Design module .................................................. 40

5. A typical subcritical target N-Factor distribution calculated by the Target Pressure

Design module .......................................................................................................... 41

6. An example of an undesirable target N-Factor distribution for both subcritical and

supercritical airfoil designs ....................................................................................... 41

7. A typical supercritical target N-Factor distribution .................................................. 42

8. An example of how an upper surface pressure distribution is modified to move a

shock back to 55% chord .......................................................................................... 42

9. The two regions that the upper surface pressures are divided into to calculate the

target pressure distribution ........................................................................................ 43

10. An illustration of how the N-Factor design method is used to calculate the new

target pressures .......................................................................................................... 44

11. The evolution of the target pressures in the recovery region .................................... 45

12. The three regions that the upper surface target pressures are divided into in order to

obtain the lift and pitching moment coefficients ....................................................... 47

13. A flowchart of the method by which the target pressures are modified to meet the

aerodynamic constraints ............................................................................................ 48

14. The upper surface pressure distribution that is used to increase the leading-edge

radius of the airfoil .................................................................................................... 49

15. The evolution of an initial lower surface target pressure distribution ...................... 49

16. A flowchart of the module for Modifying Target Pressures to Enforce Constraints 51

iv

17. The pressure distribution and shape of the NACA 641-212 airfoil at M_ = 0.10, Re

= 3 million, and c l = 0.30 .......................................................................................... 52

18. The upper and lower surface N-Factor distributions of the NACA 641-212 airfoil at

M_ = 0.10, Re = 3 million, and c l = 0.30 .................................................................. 52

19. The upper and lower surface N-Factor distributions of the redesigned NACA 641-

212 airfoil at Moo = 0.10, Re = 3 million, and c l = 0.30 ............................................ 53

20. The pressure distribution and shape of the redesigned NACA 641-212 airfoil at M_

= 0.10, Re = 3 million, and c l = 0.30 ......................................................................... 53

21. A comparison of the NACA 641-212 airfoil and the redesigned NACA 641-212

airfoil .......................................................................................................................... 54

22. The pressure distribution and shape of the NACA 1412 airfoil at M_ = 0.60, Re =

20 million, and c l = 0.40 ........................................................................................... 55

23. The upper and lower surface N-Factor distributions of the NACA 1412 airfoil at

M_ = 0.60, Re = 20 million, and c l = 0.40 ................................................................ 55

24. The upper and lower surface N-Factor distributions of the redesigned NACA 1412

airfoil at M_ = 0.60, Re = 20 million, and c l = 0.40 ................................................. 56

25. The pressure distribution and shape of the redesigned NACA 1412 airfoil at M_ =

0.60, Re = 20 million, and c I = 0.40 .......................................................................... 56

26. A comparison of the NACA 1412 airfoil and the redesigned NACA 1412 airfoil...57

27. The pressure distribution and shape of the NASA High Speed NLF-0213 airfoil at

Moo = 0.60, Re = 20 million, and c l = 0.40 ................................................................ 58

28. The upper and lower surface N-Factor distributions of the NASA High Speed NLF-

0213 airfoil at M_ = 0.60, Re = 20 million, and cI = 0.40 ........................................ 58

29. The upper and lower surface N-Factor distributions of the redesigned NLF-0213

airfoil at M_ = 0.60, Re = 20 million, and c l = 0.40 ................................................. 59

30. The pressure distribution and shape of the redesigned NLF-0213 airfoil at M_ =

0.60, Re = 20 million, and cI = 0.40 .......................................................................... 59

31. A comparison of the NASA High Speed NLF-0213 airfoil and the redesigned NLF-0213 airfoil ................................................................................................................ 60

V

32.

33.

34.

35.

36.

37.

38.

39.

A comparison of the pressure distributions of the redesigned NACA 1412 airfoil

(Figure 25) and the redesigned NASA High Speed NLF-0213 airfoil (Figure 30)..60

A comparison of the redesigned NACA 1412 airfoil (Figure 25) and the redesigned

NASA High Speed NLF-0213 airfoil (Figure 30) .................................................... 61

A comparison of the redesigned NACA 1412 airfoil and the redesigned NASAHigh Speed NLF-0213 airfoil after rotation ............................................................. 61

The pressure distribution and shape of the NASA Supercrifical SC(2)-0412 airfoil

at M_ = 0.76, Re = 10 million, and c l = 0.50 ............................................................ 62

The upper and lower surface N-Factor distributions of the NASA Supercritical

SC(2)-0412 airfoil at M_ = 0.76, Re = 10 million, and c l = 0.50 ............................. 62

The upper and lower surface N-Factor distributions of the redesigned SC(2)-0412

airfoil at M_ = 0.76, Re = 10 million, and c l = 0.50 ................................................. 63

The pressure distribution and shape of the redesigned SC(2)-0412 airfoil at M_ =

0.76, Re = 10 million, and c l = 0.50 .......................................................................... 63

A comparison of the NASA Supercritical SC(2)-0412 airfoil and the redesigned

SC(2)-0412 airfoil ..................................................................................................... 64

vi

List of Symbols

A

C

CP

C

C d

C l

Cm

Cw

d

F

G

1

M

N

P

qR

Re

rle

s

T

t

Vg

Vw

W

X

X

Y

Greek

75

5*

01(

P

g

relaxation factor for N-Factor design methodsurface curvature of airfoil, C/c

pressure coefficient, (p - Poo) / q_o

chord length (ft)

section drag coefficient, d/(qoc)

section lift coefficient, 1/(qooc)

pitching moment coefficient, (l_)//q°°c2)

phase speed of Tollmien-Schlichting waves (ft/s)

section drag (lbf/ft)

pressure gradient parameter, Equation 11

scale factor for linear scaling method

section lift (lbf/ft)

Mach number, U/ _/y. R . T

N-Factor, Equation 16

number of frequencies

static pressure (lbf/ft 2)

dynamic pressure, (1/2) p U 2

gas constant (lbf-ft/slugs.R)

free-stream Reynolds number based on chord, (poo U c)/btoo

leading-edge radius, _'le/C

chordwise distance from stagnation point, Equation 41

static temperature (R)

airfoil thickness, "i/c

local velocity (ft/s)

complex group velocity (ft/s)

velocity of Tollmien-Schlichting waves (ft/s)

weighting function for weighted averaging technique

equivalent flat plate distance, Equation 1

distance along chord of airfoil, _c/c

vertical distance of airfoil, _/c

angle of attack (degrees)

ratio of specific heats

thickness of turbulent boundary layer, Equation 8

displacement thickness of turbulent boundary layer, Equation 10

wave numbers (1/ft)

momentum thickness of turbulent boundary layer, Equation 9

relaxation factor for turbulent boundary layerrelaxation factor used to meet lift constraint

kinematic viscosity (slugs/(ft-s))

density (slugs/ft 3)

arc length along surface of airfoil (ft)

relaxation factor used to meet pitching moment constraint

vii

T,

_te

CO

Subscripta

av

cpcrit

des

e

i

J1

max

rain

F

stagT

tol

tr

u

oo

Superscripti

parameter for airfoil design method

trailing-edge angle (degrees)

wave angle (degrees)

frequency (I/s)

analysis

average

control pointcritical

designeffective

imaginaryairfoil station

lower surface

maximum

minimum

real

stagnation point

targettolerance

transition

upper surface

free-stream conditions

iteration index

sonic conditions, M = 1

dimensional quantity

°°°

VlI1

An Approach to the Constrained Design

of Natural Laminar Flow Airfoils

Bradford E. Green

Joint Institute for Advancement of Flight Sciences

The George Washington University

Abstract

A design method has been developed by which an airfoil with a substantial amount

of natural laminar flow can be designed, while maintaining other aerodynamic and geo-

metric constraints. After obtaining the initial airfoil's pressure distribution at the design

lift coefficient using an Euler solver coupled with an integral turbulent boundary layer

method, the calculations from a laminar boundary layer solver are used by a stability anal-

ysis code to obtain estimates of the transition location (using N-Factors) for the starting

airfoil. A new design method then calculates a target pressure distribution that will

increase the laminar flow toward the desired amount. An airfoil design method is then

iteratively used to design an airfoil that possesses that target pressure distribution. The

new airfoil's boundary layer stability characteristics are determined, and this iterative pro-

cess continues until an airfoil is designed that meets the laminar flow requirement and as

many of the other constraints as possible.

1.0 Introduction

Since Orville Wright first flew in December of 1903, there have been considerable

attempts to find methods to reduce the drag of airplanes. A reduction in drag means that

airplanes can operate more efficiently by using less fuel, which results in lower operating

costs and smaller, quieter engines. Also with the reduction in fuel consumption comes the

ability to produce aircraft with longer ranges and bigger payloads.

In the 1930's, it was found that longer runs of laminar flow over an airfoil resulted in a

lower profile drag and that favorable pressure gradients contributed to prolonged laminar

boundary layers (ref. 1). Using these ideas, pressure distributions having the pressure

minimum located near the position of desired transition were sought. Once the desired

pressure distribution was found, an airfoil with that pressure distribution was then derived,

using theoretical techniques such as Theodorsen's method (ref. 2), and tested. The NACA

1-6 series airfoils are examples of airfoils that were designed in this manner (ref. 3). This

was the birth of attempts to achieve long runs of natural laminar flow (NLF) to reduce air-

plane drag.

In the 1960's, a new method for creating long runs of laminar flow was utilized. Now

called laminar flow control (LFC), this method achieved laminar flow through suction

holes located at selected spanwise stations on the wing. There are two results of boundary

layer suction. First of all, boundary layer suction thins the boundary layer and lowers the

effective Reynolds number. Secondly, boundary layer suction changes the boundary layer

profiles. The changes that result contribute to boundary layer stability, which results in

longer runs of laminar flow (ref. 4). Since then, this technology has been further

researched and several airfoils using LFC have been developed (refs. 5,6).

Although the benefits of LFC are tremendous, especially in three-dimensional flows,

the physical application of an LFC system to a wing causes several problems. One prob-

lem is the increased weight that the system adds to the airplane. Since the aircraft weight

is increased, a trade-off must be made to get the true benefits of the LFC system. Another

problem of LFC is contamination from insects and icing. Often times, insect remains or

ice on the surface trip the boundary layer, reducing the efficiency of the LFC system.

Another possible problem arises if a mechanical failure occurs and the system on one

wing does not work properly. In this case, lift will be lost on the wing and its drag will

increase, causing unwanted rolling and yawing moments. If these problems with LFC

systems can be overcome, perhaps the true benefits of the technology can be experienced.

Within the past fifteen years, due to the advances of the moderu-day computer, even

more research has gone into the benefits of and the methods for achieving NLF. Although

most of this research has been applied to designing airfoils (refs. 7-11), some research has

been done on designing fuselages for NLF (refs. 12-14). The methods implemented for

designing airfoils with long runs of NLF seem to be mainly trial and error methods using

linear stability theory to assess the effect of changing the pressure distribution, although

there have been some optimization methods developed for axisymmetric airplane ele-

ments (ref. 12). Thus, the design methods employed for modifying airfoil pressure distri-

butions often required extensive knowledge and experience.

In this paper, a constrained design method is presented for modifying an airfoil's shape

such that long runs of NLF can be achieved. The design method uses an Euler solver cou-

pled with an integral turbulent boundary layer method and a laminar boundary layer solver

to calculate the velocity and temperature profiles at each airfoil station. A boundary layer

stability analysis code is then used to find the stability of the boundary layer in terms of N-

factors, which are the logarithmic amplification of the Tollmien-Schlichting (T-S) waves.

After calculating a target N-Factor distribution that forces transition to occur at the desired

location, a new method is used to calculate a target pressure distribution that is closer to

meeting the NLF constraints, while maintaining several aerodynamic and geometric con-

straints. Once the new target pressure distribution is calculated, an airfoil design method

is used to design an airfoil that has that pressure distribution. This process is iterated until

an airfoil is designed that meets the desired NLF, aerodynamic and geometric constraints.

2.0 Overview of the Airfoil Design Method

The flowchart shown in Figure 1 demonstrates the process by which the final NLF air-

foil is iteratively designed. The first module on the flowchart uses the Euler solver dis-

cussed in Section 2.1 and the turbulent boundary layer method discussed in Section 2.2 to

calculate the pressure distribution of the current airfoil. This pressure distribution is then

analyzed by the laminar boundary layer solver discussed in Section 2.3 to calculate the

boundary layer velocity and temperature profiles. The stability analysis code discussed in

Section 2.4 then uses this data to calculate the N-Factor distribution for the current airfoil

and pressure distribution. Using the current N-Factors and pressures, the Target Pressure

Design module, which is discussed in Chapter 3, calculates a target pressure distribution,

2

from which an airfoil can be designed using the flow solver and the airfoil design method

discussed in Section 2.5. However, the target pressures often need to be modified while

designing the new airfoil. These modifications are made by the Modify Target Pressures

to Enforce Constraints module, which is discussed in Chapter 4.

Chapter 5 is included to show some results of the NLF airfoil design method. Airfoils

were designed for glider, commuter and subsonic transport aircraft, which covers a wide

range of Mach numbers, Reynolds numbers, and airfoil thicknesses.

While Chapters 3 and 4 discuss the newly developed components used in this project,

the current chapter is devoted to discussing the existing codes that have been coupled

together to design new NLF airfoils. The computer used was a Silicon Graphics Indigo2

workstation with an R4000 processor; all CPU times mentioned pertain to this computer.

2.1 Euler Solver

The Euler solver used is called GAUSS2 (General purpose Approximate factorization

Upwind Scheme with Shock fitting, 2-Dimensional version), and was written by Peter

Hartwich (refs. 15-17). The code solves the two-dimensional, compressible, non-conser-

vative Euler equations on a structured mesh.

GAUSS2 uses an upwind method that constructs a finite-difference scheme based on

the theory of characteristics. Since the upwinding technique used is the non-conservative

split-coefficient matrix (SCM) method, the code is fast and efficient. The dependent vari-

ables used are the speed of sound, and the two Cartesian velocity components, all of which

are required in any upwind scheme. Entropy is chosen as the fourth dependent variable,

which reduces the energy equation to a simple convection equation.

Away from shocks, fully one-sided, second-order accurate spatial differences are used

to update the solution. The shocks are resolved using a floating shock fitting method that

allows the shock to float between two grid points. Across the shock, the Rankine-Hugo-

niot relations are explicitly used to update the solution using the speed of sound and

entropy variables.

The solution is advanced in time using a time-implicit operator containing block-tridi-

agonal matrices for the two-dimensional Euler equations. For calculating transient flows,

the second-order accurate Crank-Nicholson time differencing is used. For the steady-state

calculations used in this study, the first-order accurate Euler-backward time discretization

is used due to the quick convergence rate that results from its better damping properties.

As mentioned previously, GAUSS2 is a fast flow solver. Typically only 500 CPU sec-

onds are required to calculate the pressure distribution of the initial airfoil at the design liftcoefficient.

As shown in Figure 1, GAUSS2 must also be used each time that the airfoil design

method is used to design a new airfoil Typically, only 60 iterations through the airfoil

design method are needed to design an airfoil that possesses the desired target pressures.

Approximately 500 CPU seconds are required to complete these 60 iterations.

Required inputs for GAUSS2 include the current airfoil, the angle of attack, and the

free-stream Mach number. Outputs of the solver include the pressure distribution on the

3

surface of the airfoil, the location of any shocks on either surface, and the wave drag asso-ciated with these shocks.

2.2 Turbulent Boundary Layer Method

The compressible turbulent boundary layer method used is a modified version of the

integral method developed by Stratford and Beavers (ref. 18). The following seven equa-

tions are taken from reference 18. With M being the local Mach number, an equivalent

fiat plate distance is defined as

X = (1 +0.2M214fx( M 4M- J "l°_, 1 + 0.2M 2) dx

(1)

This distance represents the length over which a boundary layer growing on a fiat plate

would acquire the same thickness as the real boundary layer at that given location and

Mach number (ref. 19).

For free-stream Reynolds numbers Re between 1 and 10 million, the method

expresses the boundary layer thickness as

5 = 0.37X°'8°Re -°'20 , (2)

the momentum thickness as

0= 0.036(1+ 0.10M2]-°'7°X°'8°Re -°2°, (3)

and the displacement thickness as

5*= 0.046/1 + 0.80M2]°44X°'8°Re-°'2° (4)

For free-stream Reynolds numbers between 10 and 100 million, the method uses the fol-

lowing relations:

5 = 0.23X°833Re -0"167 (5)

0 = 0.022/1 + O.lOM2]-°'7°X°'833Re-°'167 (6)

5* = 0.028(1 + 0.80M2]°'44X°'833Re-°'167 (7)

In order to find expressions for the preceding equations that are a good approximation

throughout the entire range of Reynolds numbers, the equations above have been modi-

fied. The expressions used in the current application of the boundary layer method are

5 = 0.276X°'82Re -°'18 (8)

4

0_-0.027(1+010 ) °7° °82Re-°18 (9)

8* = 0.034 / 1 + 0.80M2)°'44X°'82Re -°'18 (10)

The original method of Stratford and Beavers does not include a calculation of the vis-

cous drag. As a result, the method of Squire and Young (ref. 20) is used for this purpose

and is implemented after calculating the characteristics of the turbulent boundary layer as

described above. This method extrapolates the momentum thickness at the trailing edge

of the airfoil to infinity.

In addition, the original method of Stratford and Beavers does not include a criterion

for predicting the location of turbulent separation. Therefore, the method of reference 19

is included for this reason. This method is based on the pressure gradient parameter

{" dCp'_l/2( " -6 "_-0.10

F = Cp_.X---_x ) _ 10 XRe) , (11)

where X was defined in Equation 1.

The displacement thicknesses calculated using Equation 10 are used to calculate an

effective airfoil to account for viscous effects. With j being the airfoil station, the upper

surface of the effective airfoil is calculated using the relation

i

Ye, j,u = Yj, u+K(_*)j,u + (l-K) ((_,)j,-_l, (12)

where Ye, j, u is the upper surface of the effective airfoil, y, . is the upper surface of theJ) _ l--1 _ l

current airfoil, n is a relaxation factor (typically 0.80), and (8*)j, u and (8)j, u are theupper surface displacement thicknesses of the previous and current iterations, respec-

tively. To calculate the lower surface of the effective airfoil, the relation implemented is

, i i-1Ye, j,l = Yj, 1-1_(8 )j,l- (1-K) (8*)j,l (13)

It is this effective airfoil that is analyzed by GAUSS2 in order to calculate a pressure dis-tribution.

2.3 Laminar Boundary Layer Solver

The laminar boundary layer solver used is BL3D, which was written by Venkit Iyer

(ref. 21). This code is a quick, compressible, three-dimensional solver with fourth-order

accuracy in the wall-normal direction. It is applicable to attached laminar flows where the

perfect gas and boundary layer assumptions are valid.

Since the solver is being used for two-dimensional applications, the x- and y-momen-

tum, continuity and energy equations are solved step-wise starting at the stagnation point

of the flow. The method is iterated at each station until the desired convergence is

obtained. After the solution at each station has converged, the derivative of the velocity in

the wall-normal direction at the surface is checked to see if the flow has separated. At

5

eachstation,thevelocity andtemperatureprofiles, their first andsecondderivatives,andseveralboundarylayer edgeconditionsare written out for the stability analysiscode.Thesearethe only outputsthat areneededfrom this solver. TheinputsrequiredaretheMachnumber,Reynoldsnumber,andtheairfoil with its pressuredistribution.

2.4 Stability Analysis Code

The stabilityanalysiscodeusedin this methodis the COSAL (Compressible Stability

Analysis) code written by Mujeeb Malik (ref. 22). In order to obtain the stability proper-

ties of three-dimensional compressible boundary layers, COSAL solves the eigenvalue

problem by solving an eighth-order system of differential equations. For a two-dimen-

sional case, the problem reduces to a sixth-order system of equations.

Small disturbance theory was used to obtain the basic equations for the linear stability

analysis of parallel-flow compressible boundary layers. A set of five ordinary differential

equations result from the compressible Navier-Stokes equations and small disturbance

theory. These equations are composed of one first-order continuity equation, three sec-

ond-order momentum equations, and one second-order energy equation. This system can

be reduced to a set of eight first-order ordinary differential equations.

Using a finite difference method, COSAL solves the system of basic equations. In

order to do this, the eigenvalues are initially obtained through a global eigenvaiue search

since there are no guesses available. Then, once a guess is obtained, a quick local eigen-

value search is used to continue solving the equations.

COSAL uses temporal stability theory which assumes that the disturbances grow or

decay only with time. As a result, the frequency m is assumed complex,

CO = W r + io)i, (14)

while the wave numbers _ and _ are assumed real.

COi > 0, and decay if (.0 i < 0.

With a complex group velocity defined as

v --

The disturbances are said to grow if

(15)

an N-factor used for transition prediction is calculated using the relation

(16)

where (_ represents the arc length along a curve on the surface being analyzed.

COSAL can use four different methods to integrate to find the N-Factor at each x-loca-

tion. These methods include the envelope method, the fixed wavelength and orientation

method, the fixed wavelength and frequency method, and the fixed orientation and fre-

quency method. The method used in the current application of COSAL is the envelope

6

methodwhich requiresthe real frequencycot to be specified,andthen maximizesthegrowthrate COi with respect to the wave numbers _ and _.

Inputs to COSAL include the real frequency, an initial wave angle and the boundary

layer profile data from the laminar boundary layer solver. In this application, the most

important output from COSAL is the N-Factor distribution.

When obtaining the stability characteristics of an airfoil, usually a minimum of 10 fre-

quencies must be analyzed by COSAL. For supercritical cases, perhaps as many as 20 or

25 frequencies must be analyzed. At 200 CPU seconds per frequency, COSAL requires at

least 2000 CPU seconds, but perhaps as many as 5000 CPU seconds, for each airfoil

design iteration. This makes the use of COSAL the most expensive aspect of the method.

Approximately 70% of the time required to use this NLF airfoil design method is spent inCOSAL.

2.4.1 Calculation of Analysis N-Factor Distribution

As mentioned above, the envelope method in COSAL is used to obtain the N-Factor

distribution of the particular pressure distribution being analyzed. This method requires

the real part of the frequency COr and an initial wave angle _t to be specified in order to

obtain a unique N-Factor distribution by maximizing the growth rate coi with respect tothe wave numbers _ and _. An infinite number of frequencies co occur in nature in the

F

flow over an airfoil. Therefore, in theory, in order to calculate the exact N-Factor distribu-

tion for the pressure distribution being analyzed, COSAL would have to be used to calcu-

late an N-Factor distribution for every frequency that exists in nature.

Since this is not realistic, the pressure distribution is analyzed for a range of frequen-

cies for each airfoil station j from the stagnation point (j = 1 ) to the laminar separation

point (j = k). The lower and upper frequencies of this range, denoted by cor, rain and

for, max respectively, and nf, the number of frequencies to be analyzed within this range,

are specified. From nf, the differential frequency ACOr is calculated as

ACO = for, max - ('Or, min (17)

r nf--1

This means that the initial frequencies that COSAL will analyze are cor, rain '

for, rain + At'0r' f'Or, min + 2Acor' "'" cor, max - Acor ' and cor, max"

The N-Factor at a particular x-location on the airfoil will differ with frequency. For a

constant wave angle, the N-Factor at the x-location will increase as the frequency is

increased. However, at some critical frequency cor, crit' the N-Factor will be a maximumand, as the frequency is increased beyond that, it will start to decrease. This is demon-

strated in Figure 2 in which a typical N-Factor is shown over a range of frequencies at a

point on an airfoil at a constant wave angle. In general, (.Or, crit is different for each x-location on the airfoil. As a result, even though each of the frequencies in the specified

range have been analyzed by COSAL, this does not guarantee that Oar,crit for each x-loca-tion has been determined.

Therefore, a method had to be developed that would ensure that (Dr, crit for each airfoilstation was obtained. A flowchart of this method appears as Figure 3. The process begins

7

by definingtwo variables: N i and COy.With j denoting the airfoil station, N i is the array

of maximum N-Factors, whi_e oaj is the array of critical frequencies corresp6nding to the

N-Factors of Nj. Initially, both arrays Nj and oaj are set equal to zero at every j.

When COSAL is called for the first time, the pressure distribution is analyzed at a fre-

quency of Oar,rain and the wave angle _. The N-Factor of Oar,rain at station j (defined as

Nt0,j. ) is compared to Nj at j. If N_tu,j. > N.,j then Nj is replaced with the value of No_,j,

and Oaj is replaced with Oar,rain" Then, the same process is repeated for each of the fre-

quencies in the frequency range, as shown in the flowchart. After this has been done, Nj

contains the first estimate for the array of maximum N-Factors, and COycontains the firstestimate for the critical frequencies.

However, if any frequency in array 03. is equal to oar rain' then the N-Factor at that sta-• J ,

tlon j (found in array N.) is not really the maximum Using the data in Figure 2 as anj

example, if the specified frequency range was 30000 Hz to 40000 Hz, then the frequency

found in Oaj for this particular station j would be 30000 Hz since the N-Factor at this sta-tion is the largest N-Factor in this range. But looking at the figure, the N-Factor at 30000

Hz is not really the maximum. As a result, frequencies below Oar,min must be analyzed in

order to reach the maximum N-Factor for station j. This is accomplished by extending

the frequency range by changing the value of Oar,rain to Oar,rain- AOar" After analyzing

this frequency, array oaj is checked to see if it is equal to Oar,rain -- AOar for any j. If it is,

then Oar, rain needs to be modified again. If Oar, min- AOar does not appear in array oaj,then no smaller frequencies need to be analyzed.

Conversely, if, after analyzing the specified range of frequencies, any frequency in

array t0.j is equal to oar, max' then a frequency larger than oar, max must be analyzed inorder to find the maximum N-Factor at that j. This can also be demonstrated by Figure 2.

If the specified frequency range was 6000 Hz to 10000 Hz, then 10000 Hz would corre-

spond to oar, max" Although the N-Factor at 10000 Hz is the largest in this range, it is not

the maximum N-Factor for this station. Larger frequencies must be analyzed. After ana-

lyzing frequency oar, max + AOar' array oaj is checked to see if Oar, max + AOar appears at

any j. If it does, then even a higher frequency must be analyzed. If it doesn't, then Nj isthe final array of maximum N-Factors. These N-Factors form the analysis N-Factor enve-

lope or distribution for this particular airfoil at the flow conditions.

2.4.2 Use of N-Factors to Estimate the Transition Location

Since COSAL uses the e N method, the N-Factors that are calculated can be used to

obtain an estimate of the transition location. When the N-Factors exceed a certain value,

Ntr, transition is estimated to occur at that point.

A wide range of estimates for the value of Ntr have been made, depending on whether

the correlation was determined from a wind tunnel experiment or an in-flight experiment.

Using wind tunnel experiments, values between nine and 11 have been predicted for Ntr

(refs. 23-24), although values as high as 13.5 have been estimated using in-flight experi-ments (refs. 10, 25).

2.5 Airfoil Design Method

The airfoil design method used is the CDISC (Constrained Direct Iterative Surface

Curvature) method of Richard Campbell (ref. 26). Before designing a new airfoil, the

method first modifies the initial target pressures to meet the desired aerodynamic and geo-

metric constraints. Upon obtaining a final target pressure distribution, the method modi-

fies the original airfoil to design a new airfoil that has a pressure distribution closer to the

target pressures.

For local Mach numbers less than 1.1, the airfoil is modified to meet the target pres-

sures based on the relation (ref. 26)

AC= ACpX/1 + C290"5 (18)

where C is the curvature of the airfoil, ACp is the difference between the target and analy-sis pressures, and x is a parameter equal to 1 for the upper surface and -1 for the lower

surface. For local Mach numbers greater than 1.1, the relation (ref. 26)

K(A%)(2 _A/_]/(l+(Nd_/2)-1"5AC- dx (19)

is initially used, where Moo is the free-stream Mach number. As the analysis pressures

approach the target pressures, Equation 18 is used with Equation 19 to converge the pres-

sures more quickly.

In addition to using Equations 18 and 19 to modify the airfoil, the angle of attack is

adjusted based on the difference between the analysis and target pressures in the leading-

edge region.

Some of the parameters constrained in this airfoil design method are the lift and pitch-

ing moment coefficients, the maximum airfoil thickness, front and rear spar thicknesses,

the leading-edge radius, and the trailing-edge angle.

In this application of the method, the target pressures are modified to meet the aerody-

namic constraints prior to using CDISC. Therefore, the target pressures are modified by

CDISC only to meet the geometric constraints, not the aerodynamic constraints. Further-

more, the airfoil design method has been modified so that only the lower surface target

pressures are changed to meet the geometric constraints. This has been done in an effort

not to disturb the amount of NLF that has been achieved on the upper surface.

3.0 Obtaining the Target Pressure Distribution

This chapter is devoted to the discussion of the Target Pressure Design module shown

in the flowchart in Figure 1. This module uses the analysis N-Factor distribution from the

stability analysis code to calculate a target N-Factor distribution that forces the boundary

layer to transition at the desired location. Then, using the analysis and target N-Factors, as

wellasthe analysispressures,anewtargetpressuredistributionis calculatedthatis closerto meetingthedesiredNLF constraints.A new airfoil can thenbedesignedusingthesetargetpressures.

The componentsof the Target Pressure Design module are shown in the flowchart in

Figure 4. Each of these components will now be discussed.

3.1 The Target N-Factor Distribution

In order to calculate a target pressure distribution, a target N-Factor distribution that

has the desired amount of NLF must be determined, as shown in the first module on the

flowchart in Figure 4.

A method has been developed for calculating a target N-Factor distribution from the

analysis N-Factors and four control points (Xcp, 1, Xc, 2, Xc , 3 and Xcp ' 4 ) specified in thestreamwise direction. The first control point, x c , 1, _s positioned at the location where the

analysis N-Factors first exceed an N-Factor levPl Ncp ' 1" In order to calculate a realistictarget N-Factor distribution, it is desired to retain the current analysis N-Factors as the tar-

get N-Factors ahead of xcp, l"

To calculate the target N-Factors aft of Xcp ' 1, the target N-Factors desired at the sec-

ond (N c , a ), third (N c , 3 ) and fourth (N c , 4 ) control points are specified The target N-P P •Factor _lstribution is calculated by drawing lines through these four points and smoothing

the curve using a polynomial fit.

Since the shape of the target N-Factor distribution is dependent upon the speed regime

for which the airfoil is being designed, the values for Ncp, Jp,1, Xc , 2, Ncp, 2, Xcp, Ncp,3' 3'

Xc , 4 and N c , 4 may vary from one design to the next Typicalvalues for these parame-P P

ters will now be discussed for subcritical and supercritical cases.

3.1.1 Suberitieal Cases

Using a transition N-Factor of 10 and allowing the flow to transition at 60% chord, a

typical target N-Factor distribution for a subcritical case is shown in Figure 5. These tar-

get N-Factors were obtained from the analysis N-Factors shown in the same figure, the

following control points,

and the desired target N-Factors,

xcp, 2

xcp, 3

xcp, 4

= 0.58

= 0.60

= 0.65

Ncp, 1 =3

Ncp, 2 : 8

Ncp, 3 = 10

Ncp,4 = 15

at these control points. Using Ncp ' 1 = 3 and the analysis N-Factors shown in the figure,

the first control point, Xcp ' 1, is located at approximately 18% chord.

10

Thereareseveralreasonsfor choosingatargetN-Factordistributionsimilar to theoneshownin this figure. Theregionof thetargetN-Factordistributionaheadof x c forms abuffer zone above which the target N-Factors are not allowed to grow so that th_ 2boundary

layer will remain laminar prior to the desired transition point at slightly off-design condi-

tions. Beyond x c , 2, the target N-Factors are allowed to grow rapidly to force transitionpIn this case, x , denotes the location of desired transition, which is at 60% chord for a

cp, ,_

transition N-Factor of 10. It is not necessary, however, for any control point to be located

exactly at the desired transition point.

The steep N-Factor gradient beyond Xcp ' 2 takes into account the idea that the transi-tion N-Factor may not be exactly 10. In reality, the transition N-Factor could be as low as

eight or as high as 15. Even if the transition N-Factor were eight, the airfoil in this case

would still have NLF to 55% chord. If the transition N-Factor were 15, then the airfoil

would have NLF to 65% chord. This indicates that the flow could actually undergo transi-

tion anywhere between 55% and 65% chord, an uncertainty in the transition location of10% chord.

Suppose that a target N-Factor distribution similar to the one in Figure 6 were used for

a subcritical airfoil design. If the transition N-Factor is bounded between eight and 15,

then the flow could undergo transition anywhere between 35% and 70% chord. As a

result, this is a less desirable target N-Factor distribution since the uncertainty in the tran-

sition location of the distribution is greater.

There is, however, a limit on how steep the target N-Factor gradient beyond Xcp ' 2 canbe. If the N-Factor gradient is too great, then the adverse pressure gradient required to

obtain the target N-Factors in that region will cause the laminar boundary layer to separate

in that region. Experience has shown that in order to avoid laminar separation the target

N-Factors should not be allowed to grow more than 10 N-Factors for every 10% chord.

3.1.2 Supercritieal Cases

In supercritical designs, it is much more difficult to find a realistic target N-Factor dis-

tribution. The target N-Factor distribution shown in Figure 5 is not realistic for supercriti-

cal airfoil designs. Since the large N-Factor gradient aft of Xcp ' 2 would cause an adverse

pressure gradient in the target pressure distribution, it is likely that a shock may form

ahead of Xcp ' 4" In the case of a supercritical airfoil design, this is unwanted since Xcp ' 4denotes the desired location of the shock.

Target N-Factor distributions similar to that seen in Figure 6 were initially used in the

design of airfoils in this supercritical regime; however, using these target N-Factor distri-

butions resulted in unrealistic target pressure distributions, which caused a shock to form

ahead of Xcp ' 4" As a result, since the analysis pressures would never converge to the tar-

get pressures in the design of an airfoil, the analysis N-Factors would never converge tothe target N-Factors.

After running many cases, a target N-Factor distribution similar to the one shown in

Figure 7 was found to be a realistic distribution for supercritical cases. This distribution

was obtained using the following control points

Xcp, 2 = 0.10

11

and

Xcp, 3 = 0.35

= 0.55Xcp, 4

/Vcp, 1 = -1

Ncp, 2 "- 0

Ncp, 3 = 7

Ncp, 4 = 8

The location of x c , 1 is at 0% chord since N c , 1 = -1 and the analysis N-Factor at 0%P pchord is 0. In this figure, the target N-Factors are allowed to grow rapidly from 10% to

35% chord, where the flow can tolerate the mild adverse pressure gradient that is required

to obtain this N-Factor distribution. Then, beyond 35% chord the N-Factors are allowed

to grow only by a small amount so that a shock will not form ahead of Xcp ' 4"

A result of using a target N-Factor distribution similar to the one shown in Figure 7 is

that the uncertainty in the transition location can be large. Since a steep N-Factor gradient

is not realistic in supercritical designs, an N-Factor distribution that accounts for the

uncertainty in the transition N-Factor is not possible since it is difficult to force the N-Fac-

tors to grow much higher than eight or nine. As a result, the design philosophy for super-

critical cases is to assume a transition N-Factor near eight. In doing this, the boundary

layer will remain laminar until the flow encounters a shock at Xcp ' 4" An unfortunate resultof this idea is that if the transition N-Factor is 13 in reality, then laminar separation willoccur at the shock.

Note how there are no large N-Factor gradients in the distribution beyond 35% chord,

which would indicate that there should not be any large changes in the pressure gradient

on the resulting airfoil. Thus, the airfoil that results should have a flat, roof-top pressure

distribution that is typical of many supercritical airfoils.

3.2 Extrapolation of Analysis N-Factors

The laminar boundary layer solver is valid only for attached, laminar boundary layers,

and is terminated once the laminar boundary layer separates. As a result, the boundary

layer solver only supplies data to the stability analysis code as long as the flow is attached.

This in turn means that the stability analysis code can only calculate N-Factors until the

boundary layer separates.

Suppose that the NLF constraint requires that the flow be attached to 60% chord, but

the current airfoil's boundary layer separates at 30% chord. This means that the stability

analysis code can only calculate N-Factors to 30% chord, although a target N-Factor dis-

tribution will be specified to 60% chord. The N-Factor design method requires analysis

and target N-Factors at every station ahead of the fourth control point, which is located at

60% chord in this case. This indicates that the analysis N-Factors must be extended

through the separated flow so that the N-Factor design method discussed in Section 3.3.1

can be used. As a result, the next module in the flowchart shown in Figure 4 is used for

this purpose.

12

The methodthatis usedto performtheextrapolationdependsonwhethertheflow issubcriticalor supercritical.For subcriticalcases,this is accomplishedby linearextrapola-tion basedon the analysispressuredistribution. With stationj = k being the current

location of laminar separation and j = l being the location of the fourth control point, theN-Factors between k + 1 and 1 can be calculated as

Nj-1 -Nj-2

Nj "- Nj_ 1 q- G,a,j-l,u--G,a,j-2, u (G'a'j'u--G'a'j-l'u) (20)

where N. denotes the current analysis N-Factor distribution and Cp, a,j, u denotes the cur-rent analysis pressure distribution. From experience, for stability of the N-Factor design

method the N-Factors calculated by Equation 20 are restricted as follows:

Nj < Nj_ 1 + 150 (Xj-Xj_ 1) (21)

Nj >_0 (22)

For supercritical cases, a different approach is taken. In supercritical flows, it is

unlikely that the laminar boundary layer will remain attached through the shock. More-

over, a shock wave is a discontinuity in the pressures and would cause difficulties for the

N-Factor design method that is used to design a target pressure distribution. Therefore,

calculating N-Factors aft of a shock wave, even if the flow remained attached through the

shock, is unnecessary. As a result, in the case of supercritical designs, the laminar bound-

ary layer solver is terminated at the station upstream of the shock wave.

In the case where a shock wave exists ahead of the fourth control point, it is necessary

to move the shock aft to the desired location, which is at the fourth control point, so that

the N-Factor design method can be used. This is done by imposing a fiat roof-top pressure

distribution between the current shock location and the fourth control point. If the current

shock is at station j = k and the new shock is to be located at j = l (the station repre-

senting the fourth control point), then the current shock's upstream pressure coefficient is

maintained aft to station l. That is, the analysis pressures between k + 1 and l are rewrit-ten as

i + 1 = C i (23), a, j, u p, a, k, u

are the original analysis pres-i+1

where C , a,], u are the new analysis pressures and C isures. T_is is demonstrated in Figure 8. p, a,j, u

Also in this case, analysis N-Factors only exist ahead of station k since the laminar

boundary layer solver is terminated at k due to the shock wave. This indicates a need to

extrapolate the analysis N-Factors aft to station 1 so that the N-Factor design method can

be used to calculate a target pressure distribution. Since the pressure coefficients between

k + 1 and l are the same, the extrapolation scheme described in Equation 20 cannot be

used to calculate a new analysis N-Factor distribution. Therefore, it is assumed that the N-

Factors grow only with x, and hence can be written as

13

Nj-1 -Nj-2

Nj = Nj_ 1 q- (xj-Xj_l) (24)xj_ 1 -xj_ 2

between stations k + 1 and l.

It is possible, however, that the laminar boundary layer will separate before the shock.

In this case, prior to using Equation 24, Equations 20-22 must be used between stations

m + 1 and k, where m is the location of laminar separation and k is the location of theshock.

3.3 The Upper Surface Target Pressures

Now that the analysis and target N-Factor distributions are known, a target pressure

distribution can be calculated so that the CDISC airfoil design method can be used to

design a new airfoil. The first step in this process is to calculate the upper surface target

pressure distribution. In order to do this, the upper surface pressures are divided into two

important regions, as shown in Figure 9. In the first region, a new N-Factor design

method is used to calculate the target pressures based on the difference between the target

and analysis N-Factors. The pressures in the second region are calculated from the recov-

ery pressures of the first airfoil that was analyzed. The specifics of how the pressures are

determined in these two regions will now be discussed.

3.3.1 The N-Factor Design Method

The next module on the flowchart of Figure 4 is used to calculate the upper surface tar-

get pressures in the region that extends from the stagnation point of the airfoil to the fourth

control point. This is the region where the current pressure coefficients are modified to

move the current N-Factor distribution towards the target N-Factor distribution that was

prescribed earlier. In order to do this, a new method had to be developed.

In order to achieve the desired amount of natural laminar flow, a method was devel-

oped that possessed the following properties:

1. If the current N-Factor at a given airfoil station was larger/smaller than the

target N-Factor at that station, then the pressure would have to become

more negative/positive at that station in order to decrease/increase the cur-rent N-Factor.

2. The N-Factor at a given point on the airfoil would be changed by modify-

ing only the pressure coefficient at that point. (This local change in pres-sure coefficient would have an effect on the N-Factors downstream of the

current airfoil station, but not the N-Factors upstream since the boundary

layer equations are parabolic.)

3. The design method must produce a smooth and continuous pressure distri-

bution between the stagnation point and the fourth control point.

14

With theseideasin mind, thismethod,calledtheN-Factordesignmethod,wasdevel-opedsuchthatthechangein pressurecoefficientrequiredat agivenairfoil stationj would

be governed by the linear relation

ACp, j, u = AANj (25)

where

A%,j,u = %,T,j,u-%,a,j,u (26)

ANj = NT, j - Nj (27)

In these equations, A is a relaxation factor (typically 0.012), Cp, T,j, u are the new upper

surface target pressures, Cp, a,j, u are the current upper surface analysis pressures, NT, jare the target N-Factors, and N. are the current analysis N-Factors.1

Note that Equation 25 satisfies condition 1 above in that for a positive ANj, A%,j, u is

also positive. Conversely, for a negative ANj, A%,j, u is also negative.

Now in order to satisfy condition 2 above, the N-Factors downstream of station j must

be corrected for the change in pressure that has been imposed. This is accomplished by

assuming that each N-Factor downstream of station j is increased or decreased by as

much as the N-Factor at station j was changed. Effectively, then, ANj is added to each N-Factor downstream of j. Condition 3 above requires that the pressure distribution be both

smooth and continuous. In order to maintain a continuous pressure distribution, the

change in pressure applied to station j is also applied to each station downstream of j. In

doing this, the pressure distribution remains inherently smooth.

Condition 2 indicates that each N-Factor downstream of j must be corrected for the

change in pressure coefficient at j. This perspective can be reversed. If the method is

designing at station j, all the modifications in pressures and N-Factors upstream must be

taken into account at station j. By solving Equation 25 for Cp, T,j, u and correcting forupstream effects as mentioned above, the new relation

Cp, T,j, u = (%,a,j,u+A%,j_l,u) +A(NT, j- (Nj+ANj_I) ) (28)

is obtained. With j = 1 corresponding to the stagnation point, this relation is valid from

j = 2 to j = l, the location of the fourth control point. The boundary conditions at the

stagnation point are

Cp = = C, T, 1, u %, a, 1, u p, stag (29)

NT, 1 = N a = 0 (30)

Using COSAL as the stability analysis code, Equation 30 is automatically satisfied as long

as the target N-Factor at the stagnation point is set equal to zero.

Figure 10 illustrates how the N-Factor design method is used to calculate the new tar-

get pressures. The upper-most plot in Figure 10 shows sample analysis and target N-Fac-

15

tor distributions. The change in N-Factor, AN, that is required at a grid point is calculated

as the difference between the target and analysis N-Factors at that point. The ACn that is

necessary to make this change in N-Factor is then calculated using Equation 25. _nen, as

shown in the middle plot in Figure 10, the pressure coefficient at the current grid point and

each grid point downstream is changed by ACp The analysis N-Factors at the currentgrid point and each one downstream are changed by AN to obtain a modified analysis N-

Factor distribution. This is illustrated in the lower-most plot of Figure 10. These plots

demonstrate how the N-Factor design method is used at one grid point. The same

approach is used at each of the grid points over which a target N-Factor distribution is

specified.

As mentioned above, the parameter A used in Equation 25 is a relaxation factor in the

design process. Using COSAL as the stability analysis code, a typical value for A is

0.012. Although it is not recommended for stability purposes that A be increased above

0.018, A may be decreased if more stability is desired. In addition, the value of A does

not seem to be affected by flow parameters like Mach number or Reynolds number.

Perhaps some physical significance can be found in the relationship of Equation 25.

As the Tollmien-Schlichting waves propagate downstream, their velocity is

v w = c w + U, (31)

where c w is the phase velocity of the waves and U is the local velocity of the flow. If the

flow is being accelerated (i.e., the velocity U is increasing), then the speed of the wave,

v w , is also increasing. As v w increases, the T-S waves spread out and their amplitudes

decrease, which promotes boundary layer stability. On the contrary, if the flow is being

decelerated (i.e., the velocity U is decreasing), then the speed of the wave is also decreas-

ing. As v w decreases, the T-S waves bunch-up and their amplitudes increase, which

increases the instability of the boundary layer. This indicates that there is a negative pro-

portional relationship between a change in the amplitude of the T-S waves and a change in

the local flow velocity. Since the N-Factor is merely the logarithmic growth of the ampli-

tude of the T-S waves, then there is also a negative proportional relationship between a

change in N-Factor and a change in the local flow velocity.

There is also a negative inverse relationship between a change in the local pressure

and a change in the local flow velocity• As a result, there is a direct relationship between a

change in the local pressure and a change in N-Factor, as proposed in Equation 25.

3.3.2 The Pressure Recovery Region

The next module on the flowchart of Figure 4 is used to calculate the upper surface tar-

get pressures in the recovery region. The pressure recovery region is composed of the

pressures between station j = l and the trailing edge of the airfoil (j = n ), as shown in

Figure 9. These pressures are formed by modifying the recovery pressures of the first air-

foil that was analyzed. C,., 0 denotes the upper surface pressure coefficients of the first• . pJ ......airfoil that was analyzed, with j denoting the airfoil station. From this pressure recovery,

two intermediate pressure distributions are formed and used to determine the final target

pressures in the recovery region. Figure 11 shows this process.

16

The first intermediate recovery pressure distribution is determined by linearly scaling

(see Appendix A, page 65) C,., 0, as shown in Figure 11 (a). The target pressure coeffi-pJcient at j = l was already determined by the N-Factor design method described in the

previous section. Therefore, Cp, j, 0 must be linearly scaled so that its pressure coefficientat station l is the same as the pressure coefficient at station 1 given by the N-Factor design

method. As a result, the first intermediate recovery distribution is given by

Cp...2,T. l,._u_f p, n....2, 0 O) + C

%,j, 1 = Cp, l,O_%,n, 0 (%,j,O-%,n, p,n,O

(32)

Equation 32 is valid from j = l to j = n.

The second intermediate recovery distribution is obtained by adding a linear loading

distribution onto C . ,, as shown in Figure 11 (b). This loading distribution is added inp,j, u

order to match the design pitching moment coefficient constraint. The process by which

this is done will be described in Section 3.4.2. The second intermediate recovery distribu-

tion is then given as

--X.

Cp, j, 2 = Cp, j, o + -----2:C (33)0.30 p, 0.7c

where Cp, 07c denotes the magnitude of the loading distribution at 70% chord. This equa-tion is also valid from j = I to j = n.

Now, the final target pressures of the recovery region are obtained by taking a

weighted average (see Appendix B, page 67) of the first and second intermediate recover-

ies, as shown in Figure 11 (c). In doing so, the final target pressures are given by

X n --.______Xj . _ -- X l

%,T,j,u = Xn_X,%,,,1 +X-__X/%,j, 2(34)

Being valid from j = l to j = n, this expression allows Cp, T,j, u to retain the value of

C,j, 1 at j = l and the value of Cp,j, 2 at j = n.

The procedure described in this section is only valid for subcritical flows. For cases

where there is a shock wave on the upper surface (at j = l by design), the same process is

used to determine the pressure recovery region except that everywhere an "l" appears in

the equations, "/+ 1" should be put in its place. As a result, then, the equations are only

valid from j = l + 1 to j = n for supercritical cases.

In order to use these equations from j = l + 1 to j = n, the target pressure coefficient

on the downstream side of the shock (j = l + 1 ) must be calculated using the free-stream

Mach number Moo. The relation (ref. 27)

Cp - 1. _'24M -1 (35)

17

is usefulin doingthis. SolvingEquation35for M, the Mach number on the upstream side

of the shock is calculated by

l(M1-- 1/3.5(36)

Using normal shock theory (ref. 27), the Mach number on the downstream side of the

shock is then calculated by

!_ 0 11J2M 2 = (1.4M_1_0.2 / (37)

Using Equation 35, the target pressure coefficient on the downstream side of the shock canthen be calculated as

2/(1+02 2 351/c ,2 .zM2 j

However, because of curvature of the airfoil, the pressure coefficient calculated by

Equation 38 is not totally correct. To account for curvature effects, the target pressure

coefficient on the downstream side of the shock is given as

1

%,T,l+l,u -" E(% * +%,2 ) (39)

where C * is the sonic pressure coefficient. The process previously discussed can now be

used to d°etermine the remainder of the target pressures in the recovery region.

3.4 Meeting the Aerodynamic Constraints

Once this preliminary target pressure distribution is determined, the next module on

the flowchart of Figure 4 is used to modify the upper surface target pressures to meet the

aerodynamic constraints, which are the lift and pitching moment coefficients. A process

was developed by which the upper surface target pressures could be modified to meet

these constraints, while changing the N-Factor distribution as little as possible so that theNLF would not be disturbed.

To match these constraints, the upper surface target pressure distribution is divided

into three segments. Figure 12 shows a typical upper surface target pressure and target N-

Factor distribution, with the pressure distribution being divided into the following three

regions:

1. The leading-edge region. This region extends from the stagnation point (j = 1 ) to the

last point where the target N-Factors are zero (j = m ).

2. The center region. This region extends from airfoil station m to the fourth control

point (j = l).

18

3. The pressure recovery region. This region extends from station l to the trailing edge

(j=n).

The methods by which the target pressures are modified to meet these aerodynamic

constraints are discussed in the next two sections. A flowchart showing the procedures of

the sections appears as Figure 13.

3.4.1 Lift Coefficient

Since the extent of NLF is dependent upon the pressure gradient, a method for modify-

ing the upper surface target pressures to match the lift coefficient was developed that

would maintain the current pressure gradient through the region where the N-Factors are

increasing most rapidly. The center region is the region where the N-Factors are most

important because they are growing fastest. As a result, it is desired to shift each pressure

coefficient in this region by the same amount in order to preserve the current pressure gra-

dient. But, if these pressures are shifted, then the pressures in the leading-edge and pres-

sure recovery regions must also be manipulated so that the pressure distribution remains

smooth. In the leading-edge region, the pressures are linearly scaled so that they are con-

tinuous at j = m, the beginning of the center region.

• i

In order to determine how much the current target pressures Co, T,j, u are to be modi-

fied, it is first assumed that the center region is going to be shifted by ACp, c" This wouldcause a change in lift coefficient of

ACl, = A%,c c ( Sl- Sm I---Sn (40)

where sj represents the chordwise distance from the stagnation point (j = 1 ) defined as

J

= Z Xk--Xk-1 (41)

k=2

If the pressures in the center region are shifted by ACp, c, then the pressures in the lead-ing-edge and pressure recovery regions must also be changed, as will be discussed

momentarily. The change in lift coefficient in the leading-edge and pressure recovery

regions may be approximated by a linear loading variation along the corresponding arc

length. Therefore,

ACl'le -" 2 c(42)

( \

S n -- S l ]1AACl, pr =-- _ Cp, c_ s n )|

(43)

As a result, the total change in Ac I for these changes in pressure would be the sum of

Equations 40, 42, and 43, which results in

19

Ac l = ACp, (1Gn st-sin l Sn -s/ICt2Sn + --'Sn +72 7 n )

(44)

Simplifying and solving Equation 44 for ACp, c, the amount that the target pressures inthe center pressure region must be shifted by is

2SnAC lAC - (45)

p, c SI _ Sm q_ Sn

For stability reasons, this equation can be rewritten to include a relaxation factor. In addi-

tion, the change in lift coefficient that is needed is actually the difference between the

design lift coefficient and the lift coefficient of the current target pressures. So, with a

relaxation factor _, (typically 50%), Equation 45 becomes

2_'Sn (Cl, des - Cl)AC = (46)

p, c S l -- S m -t- S n

Thus, after calculating ACp, c, the new target pressures over the upper surface fromj = m toj = lbecome

i+ 1 = C i + ACp, (47),T,j,u p,T,j,u c

From j = 1 to j = m - 1, the upper surface target pressures become

C i + 1 : p, T, m, u - Cp, T, 1, u C i - C i C i (48)p,T,j,u ci Ci p,T,j,u p,T,l,u + p,T,l,u

p, T, m, u p, T, l, u

The region between j = l + 1 and j = n is the pressure recovery region. Therefore, the

method originally used to obtain the oressures in the recovery region (described in. . _i _-1 _

Section 3.3.2) is again used to obtain Cp, T,j, u trom j = 1 to j = n.

As shown in Figure 13, this process is then repeated until

]Cl, des--Cl] <- Ol, to l (49)

where Cl, tol is the desired tolerance for the lift coefficient (typically 0.01).

3.4.2 Pitching Moment Coefficient

Once the design lift coefficient has been achieved, a modification to the pressures in

the recovery region is made in an attempt to match the design pitching moment coeffi-

cient. The method by which the target pressures in the recovery region are determined is

described in Section 3.3.2. In that section, a linear loading distribution with magnitude

Cp, 0.7c at 70% chord was included in Equation 33 for the second intermediate pressurerecovery. This exists for the sole purpose of achieving the design pitching moment coeffi-cient.

20

In order to meet the pitching moment constraint, lift is transferred to or from the pres-

sure recovery region by changing the amount of lift in the loading distribution. This is

accomplished by modifying Cp, 0.7c by adding ACp, 0.7c" The change in lift coefficient of

the target pressures resulting from adding ACp, 0.7c to Cp, 0.7c is

1

ACl = -_2 ( 1 - xl) ACp, 0.7c (50)

where x l represents the location of the fourth control point. This change in lift coefficient

causes a change in pitching moment coefficient

1 -x/) (51)ACrn = -Ac l x l- 0.25 +

Using Equation 50, Equation 51 can be simplified to

Acm = _2(1 - xl) _x t + ACp, O.7c

Simplifying and solving for ACp, 0.7c'

A%, O.7c =

24Acm

( 1 - Xl) (8x l + 1)(53)

Since the change in pitching moment required is actually the difference between Cm, des

and cm , Acm can be replaced with Cm, des - Cm" In addition, Equation 53 can be written toinclude a relaxation factor _ (typically 50%). As a result, the final expression for

ACp, 0.7c is

24g ( Cm, des -- Cm) (54)ACp'0"7c = (1--Xl) (8X/+ 1)

Therefore, the new value of Cp, 0.7c is

c +l,0.7c = Up, 0.7c + A%, 0.7c (55)

i .....

where C is the original value Using this new value of C , the target pressuresp; 0.7c " ,0.7c

in the recovery region can be recalculated using the method of _ection 3.3.2.

After changing the pressures in the recovery region to move cm towards Cm, des' thelift coefficient of the target pressure distribution was changed. As a result, the method of

Section 3.4.1 is used to modify the target pressures to once again achieve the design liftcoefficient.

This process is repeated until

Cl, des -- Cl <- Cl, tol (56)

21

and

Cm, des -- Cm <- Cm, tol (57)

A typical value for Cm, tol is 0.01.

3.5 Adjusting the Leading-edge Radius

Since the airfoil design method uses a target pressure distribution to design a new air-

foil, the leading-edge radius of the redesigned airfoil depends on the shape of both the

upper and lower surface target pressure distributions. As a result, many different target

pressure distributions can be used to meet the leading-edge radius constraint. However, it

is desirable to use a target pressure distribution that has a reasonable amount of NLF on

the lower surface, since the N-Factor design method described in Section 3.3.1 is not used

to design the lower surface target pressures to obtain NLE

In order to do this, the pressure distribution shown in Figure 14 is used to increase the

leading-edge radius of the airfoil through modifying the leading-edge target pressures of

the upper surface. These pressures were the leading-edge pressures of an airfoil that was

redesigned starting from a NACA 641-212 airfoil. The redesigned airfoil had a large

extent of NLF on the lower surface and these upper surface pressures seemed to be a con-tributor to this.

To use these leading-edge pressures, this distribution is linearly scaled so that its pres-

sure coefficient at 30% chord is the same as that of the current target pressures. Then,

over the first 30% of the chord, a weighted averaging technique is used to calculate the

new target pressures using this distribution and the current target pressures. The pressure

coefficient at the leading edge of the distribution shown in Figure 14 is retained as the new

target pressure, while the current target pressure coefficient at 30% chord remains

unchanged.

After modifying the target pressures through this averaging, the method of Section 3.4

must be used to modify the upper surface target pressures to meet the aerodynamic con-

straints. This is shown in the next module of the flowchart of Figure 4.

3.6 The Lower Surface Target Pressures

The next module on the flowchart of Figure 4 is used to calculate the lower surface tar-

get pressure distribution. Although the N-Factor design method is not applied to design

the lower surface since the lower surface target pressures are modified to meet the geomet-

ric constraints, it is desired to obtain as much NLF on the lower surface as possible. In

order to do this, the upper surface target pressure distribution is linearly scaled and a

weighted average is taken in order to obtain the initial lower surface target pressures. The

following process is only applied on the first iteration of the method since it has been seen

occasionally to cause the target pressure distribution to change too much from one itera-

tion to the next when many constraints have been imposed. Figure 15 demonstrates the

process that is about to be discussed.

The intermediate lower surface target pressures are obtained by linearly scaling the

upper surface target pressures, as demonstrated in Figure 15 (a). If Cp, T,j, u represents the

22

uppersurfacetargetpressures,then the intermediatelower surfacetargetpressuresaregivenas

,j, 1 =%,T,l,u-Cp, T,k,u-ACp, kCl, des

,T,l,u--f p, T,k,u(%,T,j,u--Cp, T,l,u ) +%,T,l,u (58)

where k denotes the airfoil station closest to 50% chord, and ACp, k is used to cause the

final target pressure distribution to have a lift coefficient of ct, des" Equation 58 is validfrom j = 1 (the stagnation point) through j = n (the trailing edge).

For the region on the lower surface forward of the fourth control point on the upper

surface, the pressures represented by Equation 58 are used for the target pressures. That

is, from j = 1 to j = l (the location of the fourth control point), the lower surface target

pressures are given by

%,T,j,l = %,j, 1 (59)

For the region aft of station l, the final lower surface target pressures are found by taking

a weighted average of the intermediate pressures and the analysis pressures of the recov-

ery region of the initial airfoil that was analyzed, which are denoted by C ,j, 0" This ispdemonstrated in Figure 15 (b). As a result, from j = l + 1 to j = nl, the lower surfacetarget pressures become

sj - s t snz - SJc

%,T,j,l -- Sn t Slf p,j,O + _ (60)_ Sn l_Sl p,j, 1

Equation 60 allows C _ . , to retain the value of C ., at j = l and the value of Cp,j, 0p, L,j, t p,], 1at the trailing edge. The final target pressure distributaon is shown in Figure 15 (c).

It should be mentioned here that these lower surface target pressures will be modified

to meet the geometric constraints. Therefore, the amount of NLF that is obtained on the

lower surface is directly a function of the geometric constraints imposed. Moreover, since

the geometric and aerodynamic constraints constantly react to the changes that the other

makes to the pressures, the amount of NLF obtained on the lower surface is also depen-

dent upon the aerodynamic and upper surface NLF constraints.

4.0 Modifying Target Pressures to Enforce Constraints

The flowchart of Figure 1 shows a module above the airfoil design method that is

labeled Modify Target Pressures to Enforce Constraints. After a target pressure distribu-

tion is calculated as described in Chapter 3, these target pressures are modified while a

new airfoil is being designed by the airfoil design method. These changes are needed in

order to enforce the desired aerodynamic and geometric constraints.

23

A flowchartof theModify TargetPressuresto EnforceConstraintsmoduleis showninFigure16. Eachof thecomponentsin this modulewill nowbediscussed.

4.1 Leading-Edge Pressures

The CDISC airfoil designmethoddirectly modifies the airfoil to meetthe leading-edgeradius constraint,without modifying the targetpressuredistribution as it doestomeetotherconstraints.As aresult,thetargetpressuresaremodifiedwithin thefirst mod-uleof Figure 16to accountfor thechangethattheairfoil designmethodmadeto thelead-ing edgeof theairfoil to meettheleading-edgeradiusconstraint.

Thismoduleis not useduntil 10iterationsthroughtheairfoil designmethodhavebeencompleted.This allows the analysispressuresto approachthe desiredtargetpressuresinthe leading-edgeregion. But, without modifying the targetpressures,theanalysispres-sureswould neverexactlymatchthetargetpressuresin this region,while still maintainingtheleading-edgeradiusconstraint.

After completingthe first 10 iterations,only the lower surfacetargetpressuresaremodified. This is doneby taking a weightedaverageaheadof 30% chordof the currentlower surfaceanalysispressuresand the current lower surfacetarget pressures.As aresult,thenewlower surfacetargetpressuresbecome

i+1 I s--J l fi s-jj fi,T,j,I = 1-- + T,j,ISk ) p,a,j,l Sk p,

(61)

where k represents the station nearest to 30% chord.

During the second 10 iterations through the airfoil design method, only the lower sur-

face target pressures are modified. This allows the upper surface analysis pressures to

approach the upper surface target pressures, which have been designed to meet the NLF

constraints. After these first 20 iterations have been completed, the upper surface analysis

pressures should closely resemble the upper surface target pressures. The new upper sur-

face target pressures are then calculated using a weighted average between the current tar-

get and analysis pressures. Therefore, the new upper surface target pressures can becalculated as

i+1,T,j,u = 1-- p,a,j, +--C ,T,j,u Sk p u

(62)

where k once again denotes the station nearest to 30% chord. In doing this, the upper sur-

face target pressures will usually not change much since the airfoil design method hascompleted 20 iterations.

4.2 Trailing-Edge Pressures

To meet the trailing-edge angle constraint, the airfoil is modified directly, without

changing the target pressures. Therefore, in the next module shown in Figure 16, the trail-

ing-edge target pressures also need to be modified to account for the changes that have

been made to the airfoil to meet the trailing-edge angle constraint. On the lower surface,

the new target pressures are calculated from the current target and analysis pressures using

24

a weightedaverageaft of 60%chord. On theuppersurface,thenewtargetpressuresarecalculatedfrom thecurrenttargetandanalysispressuresusinga weightedaveragedtech-niqueaft of the fourth controlpoint, which is at station I. In the form of equations, the

new upper and lower surface target pressures become

i+I _-Skc i Sn t-_ i,T,j,I -- Z S p,a,j,l + -- ---- T,j,l

Snt k Snt -- SkCp'(63)

i+1 _-Slfi Sn-_ i

,T,j,u -- Sn_Sl p,a,j,U+Sn_S"'_l%,T,j,u (64)

where k is the station nearest 60% chord on the lower surface, and 1 is the location of the

fourth control point on the upper surface. These modifications are made at the same fre-

quency that the leading-edge target pressures are modified to account for the leading-edge

radius constraint.

The next module on the flowchart of Figure 16 is used to modify the target pressures to

meet the aerodynamic constraints. This process was discussed in Section 3.4.

4.3 Releasing Constraints

After modifying the target pressures using the methods of the previous two sections,the next module on the flowchart is used to release one or more of the constraints in the

event that the design is over-constrained. In order to do this, a constraint priority list wasestablished. It is as follows:

1. Upper surface NLF

2. Section lift coefficient

3. Leading-edge radius

4. Spar thicknesses

5. Maximum airfoil thickness

6. Trailing-edge angle

7. Pitching moment coefficient

The pitching moment coefficient is the least important constraint and would be the first tobe released.

Throughout the design process, the target pressures are constantly being modified to

meet each of the design constraints. If the design is not over-constrained, the target pres-

sures will move toward a distribution that satisfies all of the constraints. It is possible,

however, that the problem is over-constrained and the target pressures do not move toward

any single distribution. It is in this case that one or more of the constraints must be

released to allow the target pressures to approach a distribution that meets the more impor-

tant design constraints.

25

The redesignedairfoil is comparedwith theNACA 641-212airfoil in Figure21. Inaddition,someof the characteristicsof theNACA 641-212andthe redesignedairfoil arecomparedin Table1. In thedesignof thisairfoil, thepitchingmomentandmaximumair-foil thicknessconstraintswerereleasedby theprocessdescribedin Section4.3. Neverthe-less,the designpitching momentcoefficient wascoincidentallyachieved. The fact thatthemaximumairfoil thicknessconstraintwas releasedimplies that it wasprobablynot arealisticconstraintgiventhedesiredfront andrearsparthicknessconstraints.

Table 1. A comparison of the design constraints and the

characteristics of the NACA 641-212 and the redesigned airfoil

at Moo = 0.10, Re = 3 million, and c l = 0.30

0_

c m

Cd

Xtr, u

Xtr,1

tmax

t at x = 0.20

t at x = 0.70

rle

NACA 641-212

1.48

-0.034

0.0044

0.48 (lam. sep.)

0.53 (lam. sep.)

0.120

0.104

0.067

0.0108

REDESIGN

0.67

-0.062

0.0032

0.66

0.51 (lam. sep.)

0.142

0.119

0.090

0.0140

CONSTRAINTS

-0.060

0.65

0.150

0.120

0.090

0.0140

Only five iterations of the method were required to design the new airfoil, which took

nearly four hours on a Silicon Graphics Indigo2 workstation with an R4000 processor.

5.2 Airfoil for a Commuter Aircraft

As the next example, the NLF airfoil design method was used to redesign the NACA

1412 airfoil at a subcrifical speed, with the flow conditions being

M_ = 0.60

Re = 20 million

These flow conditions are representative of a commuter aircraft.

The new airfoil was to have the following design characteristics:

Xtr,u = 0.60

c l = 0.40

cm = -0.080

tma x = 0.120

28

t = 0.095 at x = 0.20

t = 0.070 at x = 0.70

rle "--0.0100

For these flow conditions, the pressure distribution of the NACA 1412 airfoil at the design

lift coefficient is shown in Figure 22. In addition, the upper and lower surface N-Factor

distributions for this airfoil at these conditions are shown in Figure 23. In this figure, anN-Factor of 10 is used to determine where transition from laminar to turbulent flow occurs

in the boundary layer.

After calculating the target N-Factor distribution shown in Figure 24, the NLF airfoil

design method calculated the target pressure distribution shown in Figure 25. Using this

pressure distribution, the airfoil shown in Figure 25 was designed by the CDISC airfoil

design method. Figure 26 shows a comparison of the NACA 1412 airfoil and the rede-

signed airfoil.

Table 2 contains a comparison of some characteristics of the redesigned airfoil with

those of the NACA 1412 airfoil at the design flow conditions and lift coefficient. The

design constraints are also shown in this table for comparison. With the design method

imposing the tolerances specified in Equations 67-71, the redesigned airfoil meets nearly

all of the design constraints within the specified tolerances. Table 2 shows a 24 count

reduction in drag due to the extent of NLF that was achieved on both surfaces.

It took only six hours to complete the five iterations of the method that were required

to redesign this airfoil. Approximately 20% of this time was associated with the Euler

solver and CDISC airfoil design method, while 80% of this time was required by the sta-

bility analysis code.

To show that the final airfoil is nearly independent of the starting airfoil, the NASA

High Speed NLF-0213 airfoil (ref. 8) was redesigned for the same flow conditions and

design constraints as for the redesign of the NACA 1412 airfoil.

The pressure distribution of the NASA High Speed NLF-0213 airfoil at M = 0.60,

Re = 20 million, and c l = 0.40 is shown in Figure 27. The N-Factor envelopes for this

pressure distribution and these flow conditions are shown in Figure 28.

After 13 iterations of the NLF airfoil design method, the NASA High Speed NLF-

0213 airfoil was successfully redesigned to meet nearly all of the imposed constraints.

The analysis and target N-Factor distributions of the redesigned airfoil are shown in

Figure 29, while the pressure distribution of the redesigned airfoil is shown in Figure 30.

In Figure 31, the redesigned airfoil is compared with the NASA High Speed NLF-0213airfoil.

Figure 32 shows a comparison of the pressure distributions on the redesigned NASA

High Speed NLF-0213 airfoil and on the redesigned NACA 1412 airfoil. Although they

are not exactly the same, the pressure distributions do have similar shapes. The main rea-

son that these pressure distributions are not identical is because the upper surface leading-

edge pressures of the starting airfoils were very different.

29

Table 2. A comparison of the design constraints and the

characteristics of the NACA 1412 and the redesigned airfoil

at Moo = 0.60, Re = 20 million, and c l = 0.40

Cm

Cd

Xtr, u

Xtr,1

tmax

t at x = 0.20

t at x = 0.70

rlc

NACA 1412

1.78

-0.027

0.0054

0.21

0.30

0.120

0.115

0.073

0.0156

REDESIGN

0.87

-0.074

0.0030

0.59

0.48 (lam. sep.)

0.120

0.096

0.068

0.0100

CONSTRAINTS

-0.080

0.60

0.120

0.095

0.070

0.0100

Table 3 was constructed to show the similarities between the characteristics of the two

redesigned airfoils. With the exception of the angle of attack required at the design condi-

tion and the pitching moment coefficient, both airfoils appear to have identical characteris-

tics. If the pitching moment tolerance Cm, tol specified in Equation 68 had been reduced,perhaps the pitching moment coefficients of the two airfoils would be more similar.

In addition, the final target N-Factor envelopes are different, even though they both

force transition to occur near 60% chord (see Figures 24 and 28). This may have also

been due to the difference in the leading-edge pressures between the two starting airfoils.

Figure 33 shows a comparison of the two redesigned airfoils. They appear to be very

different. This is the case since the airfoil design method maintains the trailing-edge ordi-

nates of the starting airfoil throughout the design process. As a result, to better compare

the airfoils, the redesigned NASA High Speed NLF-0213 airfoil was rotated to match the

average of the trailing-edge ordinates of the redesigned NACA 1412 airfoil. Figure 34

compares the rotated NASA High Speed NLF-0213 redesigned airfoil with the redesigned

NACA 1412 airfoil. In this figure, the similarities between these two airfoils can be more

easily seen.

In this plot, it appears that the two airfoils have a different leading-edge radius. The

leading-edge radius is calculated by fitting a polynomial through the five points that com-

prise the leading-edge, with these five points being ahead of 0.5% chord. In this region,

the airfoils match very well, but then become different in the region between 2% and 20%chord.

It should also be mentioned here that a change in _ needs to be made to account for

rotating the airfoil. To do this, a would have to be increased by 0.80 ° , which would

increase the a of the redesigned NASA High Speed NLF-0213 airfoil at the design condi-

30

tion from -0.18° to 0.62° . This bettercompareswith the angleof attackat the designcondition of the NACA 1412 redesigned airfoil, which was 0.87 ° .

Table 3. A comparison of the design constraints and the

characteristics of the NASA High Speed NLF-0213 and the two

redesigned airfoils at Mo_ = 0.60, Re = 20 million, and c I = 0.40

(X

Cm

Cd

Xtr, u

Xtr,1

tmax

tatx =0.20

t at x = 0.70

rle

NLF-0213

1.03

-0.014

0.0042

0.30

0.69 (lam. sep.)

0.132

0.110

0.092

0.0095

NLF-0213

REDESIGN

-0.18

-0.081

0.0030

0.59

0.48 (lam. sep.)

0.120

0.096

0.068

0.0100

NACA 1412

REDESIGN

0.87

-0.074

0.0030

0.59

0.48 (lam. sep.)

0.120

0.096

0.068

0.0100

CONSTR.

-0.080

0.60

0.120

0.095

0.070

0.0100

5.3 Airfoil for a Subsonic Transport Aircraft

As a final example, the NASA Supercritical SC(2)-0412 airfoil (ref. 28) was rede-

signed for the following flow conditions and constraints:

Mo_ = 0.76

Re = 10 million

Xtr,u = 0.55

c l = 0.50

cm = -0.100

tmax = 0.110

t = 0.100 atx = 0.20

t = 0.065 at x = 0.70

rle = 0.0150

A successful supercritical design is much more difficult to achieve than a subcritical

design. With supercritical designs, if the specified target N-Factor distribution is not real-

istic, then it is not possible to design an airfoil that has the desired target pressure distribu-

tion, as was discussed in Section 3.1.2. As a result, trying to find a realistic target N-

31

Factordistributionfor a specificsupercriticalcaseis often a very tediousprocess.Thiswasthecasein thisdesign.

Thepressuredistribution of the NASA SupercriticalSC(2)-0412airfoil for the givenflow conditionsanddesignlift coefficientisshownin Figure35. A shockis presenton theuppersurfacenear40% chord. The upperand lower surfaceN-Factordistributionsforthispressuredistributionareshownin Figure36. UsingatransitionN-Factorof eightforthe reasonsdiscussedin Section3.1.2, laminar separationoccurredat the shockon theuppersurface,while laminarseparationoccurredat50%chordon thelower surface.

Usingthe NLF airfoil designmethod,anew airfoil wasdesignedin eight iterations.Theupperandlower surfaceanalysisN-Factorsareplottedwith theuppersurfacetargetN-Factorsin Figure37. Using the targetN-Factordistribution shownin Figure37, thepressuredistributionshownin Figure38wascalculated.Noticehow the shockhasbeenmovedaft to 55%chord,andappearsto bemuchweaker.Usingtheairfoil designmethod,the airfoil shownin this figure wasthencalculated. Theredesignedairfoil is comparedwith thestartingairfoil in Figure39.

A comparisonof the characteristicsof the NASA SupercriticalSC(2)-0412and theredesignedairfoil areshownin Table4. With the exceptionof the maximumthicknessconstraint,thefinal airfoil meetsall of thedesignconstraintsimposed,eventhoughthepitchingmomentconstraintwasreleasedafterfive iterations.

Table 4. A comparison of the design constraints and the characteristics

of the NASA Supercritical SC(2)-0412 airfoil and the redesigned airfoil

at M_ = 0.76, Re = 10 million, and c l = 0.50

Cm

Cd (wave)

ca (total)

Xtr, u

Xtr,1

tmax

t at x = 0.20

tatx =0.70

rle

SC(2)-0412

1.20

-0.075

0.0017

0.0058

0.41 (shock)

0.53 (lam. sep.)

0.120

0.109

0.073

0.0222

REDESIGN

0.58

-0.101

0.0002

0.0044

0.55 (shock)

0.32 (lam. sep.)

0.107

0.100

0.065

0.0150

CONSTRAINTS

-0.100

0.55

0.110

0.100

0.065

0.0150

Since the laminar boundary layer separated at 32% chord on the lower surface of the

redesigned airfoil, the viscous drag was not reduced in the design process. Nevertheless,

due to the reduction in wave drag, the total drag of the redesigned airfoil was 14 counts

32

less than that of the starting airfoil. Not only is this a result of moving the shock aft while

maintaining the same lift coefficient, but it also results from allowing the N-Factors to

grow as much as possible without forcing transition until the shock.

The fact that the boundary layer remained attached on the lower surface only to 32%

chord is a result of the geometric constraints that were imposed on the airfoil. As was

mentioned previously, the lower surface target pressures are modified to meet the geomet-

ric constraints. As a result, if the front spar thickness had been reduced to 9 or 9.5%, then

perhaps the airfoil design method would not have had to work so hard to increase the pres-sures aft of 25% chord in order to meet the maximum thickness constraint. On the other

hand, if the front spar thickness had not been reduced, then increasing the desired maxi-

mum airfoil thickness to 12% would have given the same effect.

6.0 Concluding Remarks

An automated two-dimensional method has been developed for designing NLF air-

foils, while maintaining several other aerodynamic and geometric constraints. The

method has been shown to work for a range of Mach numbers, Reynolds numbers, and air-

foil thicknesses. The method has also been demonstrated for a supercritical case where a

shock wave is present on the upper surface of the airfoil.

In order to develop this NLF airfoil design method, several existing CFD codes were

coupled together. In addition, a process was developed for calculating a target N-Factor

distribution that forces transition to occur at the desired location. Using this target N-Fac-

tor distribution, as well as the current analysis N-Factors and pressures, a method was also

developed for calculating a target pressure distribution. Using this target pressure distri-

bution, the current airfoil is redesigned to obtain a new airfoil that is closer to meeting the

desired NLF, aerodynamic and geometric constraints. This method has been used to

design a number of airfoils, with results shown for glider, commuter and subsonic trans-

port applications.

One advantage of this method is that it is capable of designing an airfoil in a short

amount of time. Since an Euler solver has been coupled together with a turbulent bound-

ary layer method to calculate the pressures over the airfoil, the design time is much less

than that required for Navier-Stokes codes. As a result, a new airfoil with a large extent of

upper surface NLF can be designed in only a few hours.

In addition to the reduction in computer time required, a stability analysis code has

been used to calculate N-Factors which are correlated to the transition location. Stability

analysis methods have gained respect in the past few years and the prediction of the transi-

tion location that results is taken as being fairly accurate. In this method, the stability

analysis code is automated to calculate the N-Factor distribution by varying the frequency

of the T-S waves while assuming that the disturbances grow only in time. Since an N-Fac-

tor distribution is calculated to determine the transition location of the airfoil, a design

philosophy is presented for specifying a target N-Factor distribution for both subcfifical

and supercrifical airfoil designs. Subcritical target N-Factor distributions are specified so

that the flow does not undergo transition at slightly off-design conditions and reduces the

uncertainty of the transition location by forcing the N-Factors to grow rapidly through the

33

desiredtransition location. In supercritical designs, a target N-Factor distribution is spec-ified that forces the flow to transition before the shock so that laminar flow is not termi-nated at the shock.

In order to design a new airfoil that possesses the desired target N-Factor distribution,

an N-Factor/target pressure relationship was developed. This N-Factor design method

relates a change in N-Factor at an x-location to a change in the local pressure. In addition,

this method is independent of Mach number and Reynolds number.

Another attribute of the method is that it is capable of maintaining several aerody-

namic and geometric constraints. A method was established to meet these constraints

while also maintaining the desired amount of NLF on the upper surface of the airfoil. The

approach implemented to meet these aerodynamic and geometric constraints is new. The

method dictates that the upper surface target pressures are modified to meet only the NLF

and aerodynamic constraints while the lower surface target pressures are modified to meet

only the geometric constraints.

This method has also been shown to be robust. If enough design constraints are

imposed, the airfoil that results is largely independent of the starting airfoil. Another

advantage of this method is that the codes used have been coupled together in modular

form. This allows for other codes to be used in the place of any of the current compo-

nents. The NLF airfoil design method works efficiently and well to design new NLF air-

foils. Only by using this method could one appreciate how great it really works.

There are several possibilities for extension of this research. The method could be

applied to bodies other than airfoils and wings, with possible applications including fuse-

lages and nacelles. In addition, the method could be extended to the design of airfoils for

supersonic applications. Since large sweep angles are needed for supersonic wings, cross-

flow instabilities would be a major issue. In these cases, boundary layer suction and blow-

ing is often necessary to help reduce the crossflow disturbances. As a result, when

extending the method to include supersonic designs, the method may also have to be mod-

ified to account for suction and blowing.

7.0 Acknowledgments

This research was conducted in partial fulfillment of a Master of Science degree with

the George Washington University and was completed at NASA Langley Research Cen-ter.

1)

2)

8.0 References

Jacobs, E.N.: "Preliminary Report on Laminar-Flow Airfoils and New Methods

Adopted for Airfoil and Boundary-Layer Investigations," NACA WR L-345, June1939.

Theodorsen, T; and Garrick, I.E.: "General Potential Theory of Arbitrary Wing Sec-tions," NACA Report No. 452, 1933.

34

3) Abbott, I.H.; and von Doenhoff, A.E.: Theory_ of Wing Sections, Dover Publications,

1949, pp. 62-63, 118-120.

4) Srokowski, A.J.; and Orszag, S.A.: "Mass Flow Requirements for LFC Wing

Design," AIAA 77-1222, August 1977.

5) Kanner, H.S.: "Design of a High Subsonic Supercritical Full-Chord Laminar Flow

Airfoil with Boundary Layer Suction at High Design Lift Coefficients," Master's The-

sis, George Washington University, 1994.

6) Hanks, G.W.: "Natural Laminar Flow Airfoil Analysis and Trade Studies," NASA

CR-159029, May 1979.

7) McGhee, R.J.; Viken, J.K.; Pfenninger, W.; Beasley, W.D.; and Harvey, W.D.:

"Experimental Results for a Flapped Natural-Laminar-Flow Airfoil with High/Lift

Drag Ratio," NASA TM-85788, May 1984.

8) Sewall, W.G.; McGhee, R.J.; Viken, J.K.; Waggoner, E.G.; Walker, B.S.; and Millard,

B.F.: "Wind Tunnel Results for a High-Speed, Natural Laminar-Flow Airfoil

Designed for General Aviation Aircraft," NASA TM-87602, November 1985.

9) Somers, D.M.: "Design and Experimental Results for a Natural-Laminar-Flow Airfoil

for General Aviation Applications," NASA TP-1861, June 1981.

10) Viken, J.K.: "Aerodynamic Design Considerations and Theoretical Results for a High

Reynolds Number Natural Laminar Flow Airfoil," Master's Thesis, George Washing-

ton University, 1983.

11) Rozendaal, R.A.: "Variable Sweep Transition Flight Experiment (VSTFE)-Parametric

Pressure Distribution Boundary Layer Stability Study and Wing Glove Design Task,"

NASA CR-3992, June 1986.

12) Dodbele, S.S.: "Design Optimization of Natural Laminar Flow Bodies in Compress-

ible Flow," NASA CR-4478, December 1992.

13) Dodbele, S.S.; van Dam, C.P.; and Vijgen, P.M.H.W.: "Design of Fuselage Shapes for

Natural Laminar Flow," NASA CR-3970, March 1986.

14) Dodbele, S.S.; van Dam, C.P.; Vijgen, P.M.H.W.; and Holmes, B.J.: "Shaping of Air-

plane Fuselages for Minimum Drag," Journal of Aircraft, Vol. 24, May 1987, pp. 298-304.

15)Hartwich, P.M.: "Comparison of Coordinate-Invariant and Coordinate-Aligned

Upwinding for the Euler Equations," AIAA Journal, Vol. 32, No. 9, September 1994,

pp. 1791-1799.

16) Hartwich, P.M.: "Euler Study on Porous Transonic Airfoils with a View Toward Mul-

tipoint Design," Journal of Aircraft, Vol. 30, No. 2, March-April 1993, pp. 184-191.

17) Hartwich, P.M.: "Fresh Look at Floating Shock Fitting," AIAA Journal, Vol. 29, No.

7, July 1991, pp. 1084-1091.

35

18)Stratford,B.S.; andBeavers,G.S.: "The Calculationof the CompressibleTurbulentBoundaryLayerin anArbitrary PressureGradient- A Correlationof CertainPreviousMethods,"R. & M. No. 3207,British Aeronaut.Res.Coun.,September1959.

19)Keith, J.S.; Ferguson,D.R.; Merkle, C.L.; Heck, P.H.;and Lahti, D.J.: "AnalyticalMethodfor PredictingthePressureDistribution aboutaNacelleatTransonicSpeeds,"NASA CR-2217,March 1973.

20)Squire,H.B.; andYoung,A.D.: "The Calculationof theProfileDrag of Aerofoils," R.& M. No. 1838,British Aeronaut.Res.Coun.,1938.

21)Iyer, V.: "Computation of Three-DimensionalCompressibleBoundaryLayers toFourth-OrderAccuracyonWingsandFuselages,"NASA CR-4269,January1990.

22)Malik, M.R.: "COSAL - A Black-Box CompressibleStability Analysis CodeforTransitionPredictionin Three-Dimensional Boundary Layers," NASA CR-165925,

May 1982.

23) Malik, M.R.: "Stability Theory Applications to Laminar-Flow Control," Research in

Natural Laminar Flow and Laminar-Flow Control, NASA CR-2487, 1987, pp. 219-244.

24) Viken, J.K.: "Boundary-Layer Stability and Airfoil Design," Laminar Flow Aircraft

Certification, NASA CR-2413, 1985, pp. 1-30.

25) Horstmann, K.H.; Quast, A.; and Redeker, G.: "Flight and Windtunnel Investigations

on Boundary Layer Transition at Reynolds Numbers up to 107, '' ICAS pare 88-3.7.1,1988.

26) Campbell, R.L.: "An Approach to Constrained Aerodynamic Design with Application

to Airfoils," NASA TP-3260, November 1992.

27) Anderson, J.D.: Modem Compressible Flow: with Historical Perspective, 2nd Edi-

tion, McGraw-Hill Publishing Company, 1990, pp. 67, 278.

28) Harris, C.D.: "NASA Supercritical Airfoils - A Matrix of Family-Related Airfoils,"

NASA TP-2969, March 1990.

29) Mineck, R.E; Campbell, R.L.; and Allison, D.O.: "Application of Two Procedures for

Dual-Point Design of Transonic Airfoils," NASA TP-3466, September 1994.

36

Yes (

START)

I FLOW SOLVER

Number of prescribed iterations completed?

i No

MODIFY TARGET PRESSURES ITO ENFORCE CONSTRAINTS

)

AIRFOIL DESIGN METHOD

( STOP

LAMINAR BOUNDARY LAYER SOLVER

STABILITY ANALYSIS CODE

Yes( Method converged? )

i NoTARGET PRESSURE DESIGN

Figure 1. A flowchart of the NLF airfoil design method

37

8

7

6

N 5

4

3

2

1

00

, , , r 1 , , _ , I _ _ f _ I , , , r I , , , , I

10 20 30 40 50

m/1000 (Hz)

Figure 2. A typical variation of N-Factor (N) with frequency ((Or) atany x-location on an airfoil for a constant wave angle (_)

38

START)

( f'°r=O)r, min )

COSAL _

I Increment mr by Am r

_ _ No_ Is 0)r > for,m

No I Yes

_ Do more frequencies need to be analyzed? _ _

YesI

Change Olr,min or O_r,max II

COSAL

Figure 3. A flowchart of how COSAL is used to calculate theanalysis N-Factor distribution for a pressure distribution

39

START)

Obtain target N-Factor distribution

Extrapolate analysis N-Factors

Calculate upper surface target pressures in

the region forward of the fourth control point

Calculate upper surface target pressures in the recovery region

Modify upper surface target pressures to meet aerodynamic constraints

( )NoIs this the first iteration?

Yes

Modify upper surface target pressures to increase leading-edge radius

Modify upper surface target pressures to meet aerodynamic constraints

Calculate lower surface target pressure distribution

Modify upper surface target pressures to meet aerodynamic constraints

Figure 4. A flowchart of the Target Pressure Design module

40

2O

15

10

............ TRANS N

ANALYSIS

..... TARGET

.... N_._

V _., /

/!

5

;il0 f0.00 0.25 0.50 0.75

X

Figure 5. A typical subcritical target N-Factor distributioncalculated by the Target Pressure Design module

2O

15

10

..... TRANS N OF 8

.... TRANS N OF 15

-- TARGET

00.00 0.25 0.50 0.75

Figure 6. An example of an undesirable target N-Factor distributionfor both subcritical and supercritical airfoil designs

41

2O

15

10

............ TRANS N

-- TARGET

.... N_._

A x_=x_.,

5

0<

0.0( 0.25 0.50 0.75X

Figure 7. A typical supercritical target N-Factor distribution

-1.5 -

Cp

-1.0 -

-0.5 -

0.0

0.5

1.0

...... _7

ORIGINAL

..... NEW

4th CONTROL POINT

I T = = = f _ r = r I _ _ _ _ T _ _ _ _ I0.00 0.25 0.50 0.75 1.00

X

Figure 8. An example of how an upper surface pressuredistribution is modified to move a shock back to 55 % chord

42

Cp

-1.5

-1.0

-0.5

0.0

0.5

1.0

the region forward of the the pressurefourth control point recovery

region

X I

V 4th CONTROL POINT

-- UPPER SURFACE TARGET PRESSURES

X 1

I , _ r _ I , _ _ _ I , , , , I , , _ _ I

0.00 0.25 0.50 0.75 1.00X

Figure 9. The two regions that the upper surface pressures aredivided into to calculate the target pressure distribution

43

,'u

15

N

10

00.0

20

15

10

analysis/.

/

Cp

0.0

AN

0.5

1.0 ,,,I .... I .... I .... I .... lllifl

0.0 0.1 0.2 0.3 0.4 0.5 0.6

00.0 0.1 0.2 0.3 0.4

X

0.5 0.6

Figure 10. An illustration of how the N-Factor designmethod is used to calculate the new target pressures

44

Cp

-1.5

-1.0

-0.5

0.0

0.5

1.0 .

0.5

STARTING AIRFOIL

..... 1St INTERMEDIATE

0.6 0.7 0.8 0.9 1.0 1.1X

(a) The first intermediate recovery pressures

Cp

-1.5

-- STARTING AIRFOIL

............ LINEAR LOADING

-1.0 ......... 2nd INTERMEDIATE

-0.5

0.0

...... ...........

....'"

1.0 .... = .... _ .... i .... p .... I .... i

0.5 0.6 0.7 0.8 0.9 1.0 1.1X

(b) The second intermediate recovery pressures

Figure 11. The evolution of the target pressures in the recovery region

45

-1.5

Cp

-1.0

-0.5

0.0

0.5

1.0 -

0.5

..... 1st INTERMEDIATE

......... 2nd INTERMEDIATE

-- FINAL

I I I I I I I r I P r I I I I I I 1 I I , , , , I T I , , I

0.6 0.7 0.8 0.9 1.0 1.1X

(c) The final target pressures in the recovery region

Figure 11. Concluded

46

-1.5 4O

Cp

-1.0

-0.5

0.0

0.5

1.0 , , , X,1

-0.2

leading-edgeregion

0.0

pressure distribution

]_ the pressureXm _ X1

center re_ion _ recovery

target l_-ractors ,'

//

/

- _1 _-i-

0.2 0.4 0.6 0.8 1.0

3O

2O N

10

Figure 12. The three regions that the upper surface target pressures are

divided into in order to obtain the lift and pitching moment coefficients

47

(

START)

Calculate c1 and c m_

Does c1 meet constraint? )No

Calculate ACp, c

Modify center pressures

Modify leading-edge pressures I

Modify recovery pressures

Calculate c 1 and cm

Does cI meet constraint?

Yes

Does cm meet constraint?

No

Calculate ACp,0.7c

Modify recovery pressures

)

Figure 13. A flowchart of the method by which the targetpressures are modified to meet the aerodynamic constraints

48

-1.5

Cp

-1 .(

-0.,_

0.0

0.5

1.0 -I f _ , _ i , J , _ I _ r _ _ i _ _ _ _ I

0.00 0.10 0.20 0.30 0.40

X

Figure 14. The upper surface pressure distribution that isused to increase the leading-edge radius of the airfoil

-1.5 -

-1.0 -

-- UPPER SURFACE TARGET PRESSURES

..... INTERMEDIATE PRESSURES

Cp

-0.5

0.0 -

0,5 -

\\

\

1.0 - I , _ , _ I , , , , I , _ _ , I , _ , _ I

0.00 0.25 0.50 0.75 1.00

S

(a) Linearly scaling the upper surface target pressures

Figure 15. The evolution of an initial lower surface target pressure distribution

49

Cp

-1.5

-1.0

-0.5

0.0

0.5

1.0

......... RECOVERY PRESSURES

..... INTERMEDIATE PRESSURES

V 4th CONTROL POINT

\\

\

r- I T , , , I , , , r P , , r , I , , , , I

0.00 0.25 0.50 0.75 1.00S

(b) The lower surface recovery pressures

Cp

-1.5 -

-1.0 -

-0.5

0.0

0.5

1.0

-- UPPER SURFACE TARGET PRESSURES

..... LOWER SURFACE TARGET PRESSURES

/

I , _ _ , I _ , , _ I r _ , , I , p , , I

0.00 0.25 0.50 0.75 1.00S

(c) The initial target pressure distribution

Figure 15. Concluded

5O

START)

Calculate new target pressures in the leading-edge region

Calculate new target pressures in the trailing-edge region

Modify target pressures to meet aerodynamic constraints l

Release constraints ]

Figure 16. A flowchart of the module for ModifyingTarget Pressures to Enforce Constraints

51

-1.5

0.75

-1.0

-0.5

OOo0.25

1.0

1.5 _ t 0.00 y2.0 ' ' • , I , , t , I , , , , I ,

0.00 0.25 0.50 0.75 1.00

X

Figure 17. The pressure distribution and shape of the NACA

64 z-212 airfoil at M=_ = 0.10, Re = 3 million, and cI = 0.30

2O

15

10

............ TRANS N

UPPER SURFACE

......... LOWER SURFACE

/

/

I////

I

0.25 0.50

X

0 I , , I v [ r , , , I

0.00 0.75 1.00

Figure 18. The upper and lower surface N-Factor distributions of the

NACA 641-212 airfoil at Moo = 0.10, Re = 3 million, and c I = 0.30

52

20

N

15

10

/

//

i L

........... "f_SN

/ ./ _ UPPER SURFACE

./i ......... LOWER SURFACE

/./" • - - N TARGET

I , _._.... ;.... I _ r ,,"'_" t I , _ , , I i i i I I

0.00 0.25 0.50 0.75 1.00X

Figure 19. The upper and lower surface N-Factor distributions of theredesigned NACA 641-212 airfoil at Moo = 0.10, Re = 3 million, and c l = 0.30

-1.5

-1.0

-0.5

Cp 0.0

0.5

1.0

1.5

2.0 t _ _ _ _ I _ r _ T i _ , = _ I r r _ T I

0.00 0.25 0.50 0.75 1.00

X

0.75

0.50

0.25

0.00 y

Figure 20. The pressure distribution and shape of the redesigned

NACA 641-212 airfoil at Moo = 0.10, Re = 3 million, and c I = 0.30

53

0.10

0.05

y 0.00

-0.05

-0.10

scale 5:1

..... NACA 641.212

-- REDESIGN

I , , , i I T r _ r I _ r r _ I I I , _ I

0.00 0.25 0.50 0.75 1.00

X

Figure 21. A comparison of the NACA 641-212 airfoiland the redesigned NACA 641-212 airfoil

54

Cp 0.0 -

0.5 -

1.0 -

-1.5 -

-1.0

-0.5

1.5 _._

2.0 T I I I r [ r i 1 i I r i i I I T i r I I

0.00 0.25 0.50 0.75 1.0(

X

0.75

0.50

0.25

0.00 y

Figure 22. The pressure distribution and shape of the NACA1412 airfoil at Moo = 0.60, Re = 20 million, and c I = 0.40

20- /

. /I

/ii

15 /

I/I

/

10 ./'" ...............................................

/

ii1 II

5 /

i1111 III TRANS N

UPPER SURFACE

......... LOWER SURFACE

0 I T _ T r _ I _ r r r I _ t _ I f0.00 0.25 0.50 0.75 1.00

X

Figure 23. The upper and lower surface N-Factor distributions of theNACA 1412 airfoil at Moo = 0.60, Re = 20 million, and c I = 0.40

55

20 -

I

i I

i I

15 - /

10 ...............................................................................................................

5

/ ............ TRANS N

/ / ".,_ i -- UPPER ,SURFACE

i / ",\ / ......... LOWER SURFACE

/ / ",. .," - .... .'r,_G_i J "'... ./

0 I -t _i _ , t _\, , ,,-'/, I , _ , , I T , , , I

0.00 0.25 0.50 0.75 1.00x

Figure 24. The upper and lower surface N-Factor distributions of theredesigned NACA 1412 airfoil at Moo = 0.60, Re = 20 million, and c l = 0.40

-1.5 -

-1.0 -

-0.5

Cp 0.0-

0.5 -

2.0 I _ J , _ I v f r _ I _ _ _ _ I f _ _ r I0.00 0.25 0.50 0.75 1.00

X

0.75

0.50

0.25

0.00 y

Figure 25. The pressure distribution and shape of the redesigned

NACA 1412 airfoil at Moo = 0.60, Re = 20 million, and c l = 0.40

56

0.10

scale 5:1

0.05 S i_t _ _ _ ........ -_ ___

oooi

-0.05 ...........

..... NACA 1412

-- REDESIGN

-0.10 I _ r , _ I _ , I I I , , , T I , , , I I

0.00 0.25 0.50 0.75 1.00

X

Figure 26. A comparison of the NACA 1412 airfoiland the redesigned NACA 1412 airfoil

57

Cp

-1.5

-1.0

f

_.5

0.0

0.5

1.0

1.5

2.0 I,_

0._ 0.25 0.50

X

I _ _ _ t I

0.75 1.00

0.75

0.50

0.25

0.00 y

Figure 27. The pressure distribution and shape of the NASA High

Speed NLF-0213 airfoil at Moo = 0.60, Re = 20 million, and c t = 0.40

N

20-

15

10

5

0

............ TRANS N

/ -- UPPER SURFACE

/ ..... ._'_'_" ........ _,_ -, 7 ........ LOWER SURFACE

/

/ , .... , ....0.00 0.25 0.50 0.75 1.00

X

Figure 28. The upper and lower surface N-Factor distributions of the NASA

High Speed NLF-0213 airfoil at Moo ---0.60, Re = 20 million, and c I = 0.40

58

20

15

10

0.00

/I

I II

I

i I

I'II

/i

/............ "I'N_S N

!/ -- UPPER SURFACE

......... LOWER SURFACE

- N TARGET

I _ _ J , I _ ,_'" T T T _ _ _ _ I _ J _ r I

0.25 0.50 0.75 1.00X

Figure 29. The upper and lower surface N-Factor distributions of the

redesigned NLF-0213 airfoil at Moo = 0.60, Re = 20 million, and c I = 0.40

-1.5

-1.0

-0.5

Cp 0.0

0.5

1.0

1.5

2.0 I T i

0.00_ I f _ t _ I _ _ _ _ I _ r _ t I

0.25 0.50 0.75 1.00

X

0.75

0.50

0.25

0.00 y

Figure 30. The pressure distribution and shape of the redesigned

NLF-0213 airfoil at Moo = 0.60, Re = 20 million, and c l = 0.40

59

0.10

0.05

y 0.00

-0.05

-0.10

scale 5:1

Ii _ _- .... -_ _

, .__

\\

..... NLF-0213

-- REDESIGN

I , , i i I i _ _ i I , , , , I , , , , I0.00 0.25 0.50 0.75 1.00

X

Figure 31. A comparison of the NASA High Speed NLF-0213airfoil and the redesigned NLF-0213 airfoil

-1.0 -

Cp

-0.5 -

0.0

0.5 -

1.00.00

\

-- REDESIGN OFNACA 1412

..... REDESIGN OF NLF-0213

i i r i I i i i i J I t r I I I I T I I I

0.25 0.50 0.75 1.00X

Figure 32. A comparison of the pressure distributions of theredesigned NACA 1412 airfoil (Figure 25) and the redesigned

NASA High Speed NLF-0213 airfoil (Figure 30)

60

0.10

0.05

y 0.00

-0.05

-0.10

scale 5:1

-- REDESIGN OF NACA 1412

..... REDESIGN OF NLF-0213

' , , , , I , , , T l , , , , I , , , , I

0.00 0.25 0.50 0.75 1.00

X

Figure 33. A comparison of the redesigned NACA 1412 airfoil (Figure 25)and the redesigned NASA High Speed NLF-0213 airfoil (Figure 30)

0.10

0.05

y 0.00

-0.05

-0.10

scale 5:1

-- REDESIGN OF NACA 1412

..... ROTATED REDESIGN OF NLF-0213

I , , _ , I i i J i I , , r J I , , , i I

0.00 0.25 0.50 0.75 1.00

X

Figure 34. A comparison of the redesigned NACA 1412 airfoil andthe redesigned NASA High Speed NLF-0213 airfoil after rotation

61

-1.5 -

-1.0 -

-0.51

Cp 0.0-

0.5-

1.0-

1.5

2.00.00 0.25 0.50 0.75 1.00

x

0.75

0.50

0.25

0.00 y

Figure 35. The pressure distribution and shape of the NASA SupercriticalSC(2)-0412 airfoil at Moo =0.76, Re = 10 million, and c l = 0.50

20 -

15 -

N

10 -

/

5 - _ ............ TRANS N

-- UPPER SURFACE

/.,' ......... LOWER SURFACE

.d"

0 i . "-;_'s I n , , I I , , , , I

0.00 0.25 0.50 0.75 1.00X

Figure 36. The upper and lower surface N-Factor distributions of the NASA

Supercritical SC(2)-0412 airfoil at Moo -- 0.76, Re = 10 million, and c l = 0.50

62

20

15

10

i,0._

' ' I

0.25 0.50

X

............ TRANS N

UPPER SURFACE

......... LOWER SURFACE

..... N TARGET

0.75 1.00

Figure 37. The upper and lower surface N-Factor distributions of the

redesigned SC(2)-0412 airfoil at Moo = 0.76, Re = 10 million, and c I = 0.50

-1.5

-1.0

-0.5

Cp 0.0

0.5

1.0

1.5

2.0 I T i i ] f t T _ T I T r _ I I i T _ i I0.00 0.25 0.50 0.75 1.00

X

0.75

0.50

0.25

0.00 y

Figure 38. The pressure distribution and shape of the redesignedSC(2)-0412 airfoil at Moo = 0.76, Re = 10 million, and c l = 0.50

63

0.10

0.05

y 0.00

-0.05

-0.10

scale 5:1

..... NASA SUPERCRITICAL SC(2)-0412

-- REDESIGN

I , , _ , I , , , _ [ , , , f I _ i r , I

0.00 0.25 0.50 0.75 1.00

X

Figure 39. A comparison of the NASA Supercritical SC(2)-0412 airfoiland the redesigned SC(2)-0412 airfoil

64

Appendix A. A Linear Scaling Method

In a few cases within the NLF airfoil design method, it is desirable to calculate a new

target pressure distribution that has nearly the same shape as another pressure distribution,

but different pressure magnitudes. This can be accomplished by linearly scaling an exist-

ing pressure distribution. Consider the pressure distribution shown in Figure A.1. Sup-

pose that this pressure distribution, Cp,j, is to be linearly scaled to obtain a new targetpressure distribution. In addition, assume that the pressure coefficient at 50% chord of the

new target pressures is to be -0.30, while the leading-edge pressure coefficient is to remain

unchanged. The new target pressures can be calculated using the relation

Cp, T, j - (Cp, j-ep, 1) G+ep, 1 (A.1)

where G is the scale factor and Cp, 1

%,j. In this case,

represents the leading-edge pressure coefficient of

- 0.30 - Cp, 1G = (A.2)

where k represents the station nearest to 50% chord. Notice that when Equations A. 1 and

A.2 are used, Cp, r, 1 = Cp, 1 and Cp, T, k = -0.30 as desired. The new target pressuredistribution that results from using Equations A.1 and A.2 is shown in Figure A.2.

Cp

-1.5

-1.0

-0.5

0.0

0.5

1.0

..... Cp_

\\

\\

I , , , , I , , , , [ , , , , I r r , r I

0.00 0.25 0.50 0.75 1.00

X

Figure A.1. The pressure distribution used to calculate the new target pressures

65

Cp

-1.5

-1.0

-0.5

0.0

0.5

1.0

f

I , , , , I , , , , I , , , , I I , , r (

0.00 0.25 0.50 0.75 1.00X

Figure A.2. The new target pressure distribution

that results from using a linear scaling technique

66

Appendix B. A Weighted Averaging Technique

On several instances within the NLF airfoil design method, a weighted averaging tech-

nique is used to calculate a new target pressure distribution from two existing pressure dis-

tributions. This weighted averaging technique calculates a new target pressure

distribution using the relation

%,T,j : %%,j, 1 + (1--%)%,j, 2 (B.1)

where Wj is a weighting function, and C ,j, 1 and C,., 2 are two existing pressure distri-p pJbutions. A similar weighted averaging technique is used in reference 29.

In the NLF airfoil design method, Equation B. 1 is most useful when Wj is allowed tovary along the chord since it is usually desirable to maintain certain characteristics of each

of the two existing pressure distributions. Consider the two pressure distributions shown

in Figure B. 1. Suppose that these two distributions are to be used to calculate a new target

pressure distribution, and that the leading-edge pressures of Cp j, 1 and the trailing-edgepressures of Cp, j, 2 are to be retained as the target pressures In each of the respective

regions. If W, is allowed to vary from a value of 1 at the leading edge to a value of 0 at

the trailing edge, it can be seen from Equation B.1 that the resulting target pressures would

have the desired properties in the leading-edge and trailing-edge regions.

The simplest expression for Wj that satisfies these requirements would be a linearvariation between the leading edge and the trailing edge. That is,

Wj = 1-xj (B.2)

Figure B.2 shows the new target pressures that result from using Cp, j, 1Figure B.1 and the weighting function from Equation B.2.

and %,j, 2 from

Equation B. 1 can also be used to modify the target pressures over only a small region

of the chord. In fact, this is the only application of the technique that is used within the

NLF airfoil design method. Suppose that Cp, j, 1 represents the current target pressures

and Cp, j, 2 represents the current analysis pressures. It may be desirable to retain the tar-get pressures of C ; ahead of 60% chord (i e, W_ = 1 ahead of 60% chord), but, at the_,j, 1 " " j

same time, retain the characteristics of Cp,., 2 at the trailing edge. If station k represents

to 60% chord, then Wj Jean be varied linearly from a value of 1 at sta-the ordinate closest

tion k to a value of 0 at the trailing edge. As a result, the weighting function aft of 60%

chord can be represented by the expression

1-xj- (B.3)

Wj 1 - x k

The target pressure distribution that results from using Equation B. 1 is shown in FigureB.3.

67

Cp

-1.5

-1.0

-0.5

0.0

0.5

1.0

",,

ti "\ \

ri

ii

i

i

i

i

I

0.00 0.25 0.50 0.75 1.00X

Figure B.1. Two existing pressure distributions

Cp

-1.5

-1.0

-0.5

0.0

0.5

1.0

..... Cp.j,_

......... Cp,j,2

-- Cp,T,j

_._._._._. ......................... _..

I _ r _ _ I _ , _ _ I _ _ _ _ I _ _ t _ I

0.00 0.25 0.50 0,75 1.00X

Figure B.2. The new target pressure distribution obtained by

using a weighting function over the entire length of the chord

68

Cp

-1.5

-1.0

-0.5

0.0

0.5

1.0

..... Cr,j,1

......... Cp,j,2

-- Cp,Tj

....... -_" ,\

f._" '\

I _ _ _ _ I _ _ _ _ P _ _ _ , f _ _ , _ I

0.00 0.25 0.50 0.75 1.00X

Figure B.3. The new target pressure distribution obtained by

using a weighting function over a small region of the chord

69

Form Ap_'ovedREPORT DOCUMENTATION PAGE OMBNo.0704-0188

Puldio¢mpo_ btackmfor this cole_lon of _ Imw#mlted to am_rmgeI hou¢I_¢ mmponme,It_udiogthe Umefor mvkm_nglttstrt_-tone,mearc_knge_JrJr_dam imurcm;la#tedr_lind maintainingthe data needed,andcompletklgand revlewfnglhe cohctlon of Information.Sendo0mmentlregardingth_ burdene41tlrrmteor any othermpect Ofthismlioctlonof Information,Indudlng luggutJormfor reducing_b burden,to W_hlngton Headquartem Setvio_, Dkectomlefor tnfom_ltion Operationsand Reports,1215

_. Hk_h_y,S_e1204,_, V^22202-4._e,aMtotheO_a,ofM_emere andB.dQ_,Papo_oH(Rnd.c_o.Project(0704-01ee),Wuhl_to.,DC20e0_1. AGENCY USE ONLY (Leave bkmk) 2. REPORTDATE 3. REPORT TYPE AND DATES COVERED

February 1997 Conlzactor Report4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

An Approach to the Constrained Design of Natural Laminar Flow Airfoils

L/gffHOR(S)Bradford E. Green

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(F.S)

The George Washington UniversityJoint Inslitute for Advancement of Flight SciencesLangley Research CenterHampton, VA 23681-001

!9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space AdministrationLangley Research CenterHampton, VA 23681-0001

NCC1-24

WU 505-59-53-05

8. PERFORMING ORGANIZATIONREPORT NUMBER

10. SPONSORING/MONITORING

AGENCY RF..PORT NUMBER

NASA CR-201686

11. SUPPLEMENTARY NOTES

Langley Technical Monitor: Richard L. Campbell

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified-Unlimited

Subject Category 02Availability: NASA CASI (301) 621-0390

13. ABSTRACT (Ma_dmum 200 words)

A design method has been developed by which an airfoil with a substantial amount of natural laminar flow canbe designed, while maintaining other aerodynamic and geometric constraints. After obtaining the initial airfoil'spressure distribution at the design lift coefficient using an Euler solver coupled with an integral turbulentboundary layer method, the calculations from a laminar boundary layer solver are used by a stability analysiscode to obtain estimates of the transition location (using N-Factors) for the starting airfoil. A new design methodthen calculates a target pressure distribution that will increase the laminar flow toward the desired amount. An

airfoil design method is then iteratively used to design an airfoil that possesses that target pressure distribution.The new airfoirs boundary layer stability characteristics are determined, and this iterative process continues untilan airfoil is designed that meets the laminar flow requirement and as many of the other constraints as possible.

14. SUBJECT TERMS

Natural laminar flow; Airfoil design; Stability analysis

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATIONOF REPORT OF THIS PAGE OF ABSTRACT

Unclassified Unclassified Unclassified

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A05

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Standard Form 298 (Rev. 2-89)Prescribedby ANSI Std. Z39-18298-102